the contraction mapping principle and some … · a metric space m is said to be complete if, and...

Electronic Journal of Differential Equations, Monograph 09, 2009, (90 pages).

ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu

ftp ejde.math.txstate.edu (login: ftp)

THE CONTRACTION MAPPING PRINCIPLE AND SOMEAPPLICATIONS

ROBERT M. BROOKS, KLAUS SCHMITT

Abstract. These notes contain various versions of the contraction mappingprinciple. Several applications to existence theorems in the theories of dif-

ferential and integral equations and variational inequalities are given. Alsodiscussed are Hilbert’s projective metric and iterated function systems

Contents

Part 1. Abstract results 21. Introduction 22. Complete metric spaces 23. Contraction mappings 15

Part 2. Applications 254. Iterated function systems 255. Newton’s method 296. Hilbert’s metric 307. Integral equations 448. The implicit function theorem 579. Variational inequalities 6110. Semilinear elliptic equations 6911. A mapping theorem in Hilbert space 7312. The theorem of Cauchy-Kowalevsky 76References 85Index 88

2000 Mathematics Subject Classification. 34-02, 34A34, 34B15, 34C25, 34C27, 35A10,35J25, 35J35, 47H09, 47H10, 49J40, 58C15.Key words and phrases. Contraction mapping principle; variational inequalities;Hilbert’s projective metric; Cauchy-Kowalweski theorem; boundary value problems;

differential and integral equations.c©2009 by Robert Brooks and Klaus Schmitt.

Submitted May 2, 2009. Published May 13, 2009.

1

2 R. M. BROOKS, K.SCHMITT EJDE-2009/MON. 09

Part 1. Abstract results

1. Introduction

1.1. Theme and overview. The contraction mapping principle is one of the mostuseful tools in the study of nonlinear equations, be they algebraic equations, integralor differential equations. The principle is a fixed point theorem which guaranteesthat a contraction mapping of a complete metric space to itself has a unique fixedpoint which may be obtained as the limit of an iteration scheme defined by repeatedimages under the mapping of an arbitrary starting point in the space. As such, it is aconstructive fixed point theorem and, hence, may be implemented for the numericalcomputation of the fixed point.

Iteration schemes have been used since the antiquity of mathematics (viz., theancient schemes for computing square roots of numbers) and became particularlyuseful in Newton’s method for solving polynomial or systems of algebraic equationsand also in the Picard iteration process for solving initial value and boundary valueproblems for nonlinear ordinary differential equations (see, e.g. [58], [59]).

The principle was first stated and proved by Banach [5] for contraction mappingsin complete normed linear spaces (for the many consequences of Banach’s work see[60]). At about the same time the concept of an abstract metric space was intro-duced by Hausdorff, which then provided the general framework for the principlefor contraction mappings in a complete metric space, as was done by Caccioppoli[17] (see also [75]). It appears in the various texts on real analysis (an early onebeing, [56]).

In these notes we shall develop the contraction mapping principle in severalforms and present a host of useful applications which appear in various places inthe mathematical literature. Our purpose is to introduce the reader to severaldifferent areas of analysis where the principle has been found useful. We shall dis-cuss among others: the convergence of Newton’s method; iterated function systemsand how certain fractals are fixed points of set-valued contractions; the Perron-Frobenius theorem for positive matrices using Hilbert’s metric, and the extensionof this theorem to infinite dimensional spaces (the theorem of Krein-Rutman); thebasic existence and uniqueness theorem of the theory of ordinary differential equa-tions (the Picard-Lindelof theorem) and various related results; applications to thetheory of integral equations of Abel-Liouville type; the implicit function theorem;the basic existence and uniqueness theorem of variational inequalities and hencea Lax-Milgram type result for not necessarily symmetric quadratic forms; the ba-sic existence theorem of Cauchy-Kowalevsky for partial differential equations withanalytic terms.

These notes have been collected over several years and have, most recently, beenused as a basis for an REU seminar which has been part of the VIGRE programof our department. We want to thank here those undergraduate students whoparticipated in the seminar and gave us their valuable feedback.

2. Complete metric spaces

In this section we review briefly some very basic concepts which are part ofmost undergraduate mathematics curricula. We shall assume these as requisiteknowledge and refer to any basic text, e.g., [15], [32], [62].

EJDE-2009/MON. 09 THE CONTRACTION MAPPING PRINCIPLE 3

2.1. Metric spaces. Given a set M, a metric on M is a function (also called adistance)

d : M×M → R+ = [0,∞),that satisfies

d(x, y) = d(y, x), ∀x, y ∈ Md(x, y) = 0, if, and only if, x = y

d(x, y) ≤ d(x, z) + d(y, z), ∀x, y, z ∈ M,

(2.1)

(the last requirement is called the triangle inequality). We call the pair (M,d) ametric space (frequently we use M to represent the pair).

A sequence xn∞n=1 in M is said to converge to x ∈ M provided that

limn→∞

d(xn, x) = 0.

This we also write as

limnxn = x, or xn → x as n→∞.

We call a sequence xn∞n=1 in M a Cauchy sequence provided that for all ε > 0,there exists n0 = n0(ε), such that

d(xn, xm) ≤ ε, ∀n,m ≥ n0.

A metric space M is said to be complete if, and only if, every Cauchy sequence inM converges to a point in M.

Metric spaces form a useful-in-analysis subfamily of the family of topologicalspaces. We need to discuss some of the concepts met in studying these spaces. Wedo so, however, in the context of metric spaces rather than in the more generalsetting. The following concepts are normally met in an advanced calculus or foun-dations of analysis course. We shall simply list these concepts here and refer thereader to appropriate texts (e.g. [32] or [72]) for the formal definitions. We considera fixed metric space (M,d).

• Open balls B(x, ε) := y ∈ M : d(x, y) < ε and closed balls B[x, ε] := y ∈M : d(x, y) ≤ ε;

• open and closed subsets of M;• bounded and totally bounded sets in M;• limit point (accumulation point) of a subset of M;• the closure of a subset of M (note that the closure of an open ball is not

necessarily the closed ball);• the diameter of a set;• the notion of one set’s being dense in another;• the distance between a point and a set (and between two sets).

Suppose (M,d) is a metric space and M1 ⊂ M. If we restrict d to M1×M1, thenM1 will be a metric space having the “same” metric as M. We note the importantfact that if M is complete and M1 is a closed subset of M, then M1 is also a completemetric space (any Cauchy sequence in M1 will be a Cauchy sequence in M; henceit will converge to some point in M; since M1 is closed in M that limit must be inM1).

The notion of compactness is a crucial one. A metric space M is said to becompact provided that given any family Gα : α ∈ A of open sets whose unionis M, there is a finite subset A0 ⊂ A such that the union of Gα : α ∈ A0 is M.(To describe this situation one usually says that every open cover of M has a finite


subcover.) We may describe compactness more “analytically” as follows. Givenany sequence xn in M and a point y ∈ M; we say that y is a cluster point of thesequence xn provided that for any ε > 0 and any positive integer k, there existsn ≥ k such that xn ∈ B(y, ε). Thus in any open ball, centered at y, infinitely manyterms of the sequence xn are to be found. We then have that M is compact,provided that every sequence in M has a cluster point (in M).

In the remaining sections of this chapter we briefly list and describe some usefulexamples of metric spaces.

2.2. Normed vector spaces. Let M be a vector space over the real or complexnumbers (the scalars). A mapping ‖ · ‖ : M → R+ is called a norm provided thatthe following conditions hold:

‖x‖ = 0, if, and only if, x = 0 (∈ M)

‖αx‖ = |α|‖x‖, ∀ scalar α, ∀x ∈ M‖x+ y‖ ≤ ‖x‖+ ‖y‖, ∀x, y ∈ M.

(2.2)

If M is a vector space and ‖ · ‖ is a norm on M, then the pair (M, ‖ · ‖) is calleda normed vector space. Should no ambiguity arise we simply abbreviate this bysaying that M is a normed vector space. If M is a vector space and ‖ · ‖ is a normon M, then M becomes a metric space if we define the metric d by

d(x, y) := ‖x− y‖, ∀x, y ∈ M.

A normed vector space which is a complete metric space, with respect to the metricd defined above, is called a Banach space. Thus, a closed subset of a Banach spacemay always be regarded as a complete metric space; hence, a closed subspace of aBanach space is also a Banach space.

We pause briefly in our general discussion to put together, for future reference,a small catalogue of Banach spaces. We shall consider only real Banach spaces, thecomplex analogues being defined similarly.

In all cases the verification that these spaces are normed linear spaces is straight-forward, the verification of completeness, on the other hand usually is more difficult.Many of the examples that will be discussed later will have their setting in completemetric spaces which are subsets or subspaces of Banach spaces.

Examples of Banach spaces.

Example 2.1. (R, | · |) is a simple example of a Banach space.

Example 2.2. We fix N ∈ N (the natural numbers) and denote by RN the set

RN := x : x = (ξ1, . . . , ξN ), ξi ∈ R, i = 1, . . . , N.

There are many useful norms with which we can equip RN .

(1) For 1 ≤ p <∞ define ‖ · ‖p : RN → R+ by

‖x‖p :=( N∑i=1

|ξi|p)1/p

, x ∈ RN .

These spaces are finite dimensional lp− spaces. Frequently used norms are‖ · ‖1, and ‖ · ‖2.


(2) We define ‖ · ‖∞ : RN → R+ by

‖x‖∞ := max|ξi| : 1 ≤ i ≤ N.This norm is called the sup norm on RN .

The next example extends the example just considered to the infinite dimensionalsetting.

Example 2.3. We let

R∞ := x : x = ξi∞i=1, ξi ∈ R, i = 1, 2, . . . .Then R∞, with coordinate-wise addition and scalar multiplication, is a vector space,certain subspaces of which can be equipped with norms, with respect to which theyare complete.

(1) For 1 ≤ p <∞ define

lp := x = ξi ∈ R∞ :∑i

|ξi|p <∞.

Then lp is a subspace of R∞ and

‖x‖p :=( ∞∑i=1

|ξi|p)1/p

defines a norm with respect to which lp is complete.(2) We define

l∞ := x = ξi ∈ R∞ : supi|ξi| <∞.

and‖x‖∞ := sup

i|ξi|, x ∈ l∞.

With respect to this (sup norm) l∞ is complete.

Example 2.4. Let H be a complex (or real) vector space. An inner product on His a mapping (x, y) 7→ 〈x, y〉 (H ×H → C) which satisfies:

(1) for each z ∈ H 〈·, z〉 : H → C is a linear mapping,(2) 〈x, y〉 = 〈y, x〉 for x, y ∈ H ( 〈x, y〉 = 〈y, x〉 if H is a real vector space and

the inner product is a real valued function),(3) 〈x, x〉 ≥ 0, x ∈ H, and equality holds if, and only if, x = 0. If one defines

‖x‖ :=√〈x, x〉,

then (H, ‖ · ‖) will be a normed vector space.If it is complete, we refer to H as a Hilbert space. We note that (RN , ‖ · ‖2) and l2

are (real) Hilbert spaces.

Spaces of continuous functions are further examples of important spaces in anal-ysis. The following is a brief discussion of such spaces.

Example 2.5. We fix I = [a, b], a, b ∈ R, a < b, and k ∈ N∪0. Let K = R, or C(the reader’s choice). Define

Ck(I) := f : I → K : f, f ′, . . . , f (k), exist and are continuous on I.We note that

C0(I) := C(I) = f : I → K : f is continuous on I.


For f ∈ C(I), we define‖f‖∞ := max

x∈[a,b]|f(t)|

(the sup-on-I norm). And for f ∈ Ck(I)

‖f‖ :=k∑i=0

‖f (i)‖∞.

With the usual pointwise definitions of f + g and αf (α ∈ K) and with the normdefined as above, it follows that Ck(I) is a normed vector space. That the space isalso complete follows from the completeness of C(I) with respect to the sup-on-Inorm (see, e.g., [15]).

Another useful norm is

‖f‖∗ := supx∈I

k∑i=0

|f (i)(x)|.

which is equivalent to the norm defined above; this follows from the inequalities

‖f‖∗ ≤ ‖f‖ ≤ (k + 1)‖f‖∗, ∀f ∈ Ck(I).

Equivalent norms give us the same idea of closeness and one may, in a givenapplication, use, of equivalent norms, that which makes calculations or verificationseasier or gives us more transparent conclusions.

Example 2.6. Let Ω be an open subset of RN , and let K be as above; define

C(Ω) := C0(Ω) := f : Ω → K such that f is continuous on Ω.Let

‖f‖∞ := supx∈Ω

|f(x)|.

Since the uniform limit of a sequence of continuous functions is again continuous,it follows that the space

E := f ∈ C(Ω) : ‖f‖∞ <∞is a Banach space.

If Ω is as above and Ω′ is an open set with Ω ⊂ Ω′, we let

C(Ω) := the restriction to Ω of f ∈ C(Ω′).If Ω is bounded and f ∈ C(Ω), then ‖f‖∞ < +∞. Hence C(Ω) is a Banach space.

Example 2.7. Let Ω be an open subset of RN . Let I = (i1, . . . , iN ) be a multiindex,i.e. ik ∈ N∪ 0 (the nonnegative integers), 1 ≤ k ≤ N . We let |I| =

∑Nk=1 ik. Let

f : Ω → K. Then the partial derivative of f of order I, DIf(x), is given by

DIf(x) :=∂|I|f(x)

∂i1x1 . . . ∂iNxN,

where x = (x1, . . . , xN ). Define

Cj(Ω) := f : Ω → K such that DIf ∈ C(Ω), |I| ≤ j.Let

‖f‖j :=j∑

k=0

max|I|≤k

‖DIf‖∞.


Then, using further convergence results for families of differentiable functions itfollows that the space

E := f ∈ Cj(Ω) : ‖f‖j < +∞

is a Banach space.The space Cj(Ω) is defined in a manner similar to the space C(Ω) and if Ω is

bounded Cj(Ω) is a Banach space.

2.3. Completions. In this section we shall briefly discuss the concept of the com-pletion of a metric space and its application to completing normed vector spaces.

Theorem 2.8. If (M,d) is a metric space, then there exists a complete metricspace (M∗,d∗) and a mapping h : M → M∗ such that

(1) h is an isometry d∗(h(x), h(y)) = d(x, y), x, y ∈ M)(2) h(M) is dense in M∗.

We give a short sketch of the proof. We let C be the set of all Cauchy sequencesin M. We observe that if xn and yn are elements of M, then d(xn, yn) is aCauchy sequence (hence, convergent) sequence in R, as follows from the triangleinequality. We define dC : C × C → R, by

dC(xn, yn) := limn→∞

d(xn, yn).

The mapping dC is a pseudo-metric on C (lacking only the condition

dC(xn, yn) = 0 ⇒ xn = yn

from the definition of a metric). The relation R defined on C by

xnRyn, if, and only if, limn→∞

d(xn, yn) = 0,

or equivalently,dC(xn, yn) = 0,

is an equivalence relation on C. The space C/R, the set of all equivalence classesin C, shall be denoted by M∗. If we denote by Rxn, the class of all zn whichare R− equivalent to xn, we may define

d∗(Rxn, Ryn) := dC(xn, yn).

This defines a metric on M∗.We next connect this to M. There is a natural mapping of M to C given by

x 7→ x

(the sequence, all of whose entries are the same element x). We clearly have

dC(x, y) = d(x, y)

and thus the mapping x 7→ Rx (which we now call h) is an isometry of M to M∗.That the image h(M) is dense in M∗, follows easily from the above construction.

Remark 2.9. We observe that (M∗,d∗) is “essentially unique”. For, if (M1,d1)and (M2,d2) are completions of M with mappings h1, respectively h2, then thereexists a mapping g : M1 → M2 such that

(1) g is an isometry of M1 onto M2,(2) h2 = g h1.


The above is summarized in the following theorem, whose proof is similar to theproof of the previous result (Theorem 2.8) recalling that a norm defines a metricand where we define (using the notation of that theorem)

‖xn‖C := limn→∞

‖xn‖,

for a given Cauchy sequence, and

‖Rxn‖∗ := ‖xn‖C .

It is also clear that the set X∗ becomes a vector space by defining addition andscalar multiplication in a natural way.

Theorem 2.10. If (X, ‖ · ‖) is a normed vector space, then there exists a (essen-tially unique) complete normed vector space (a Banach space) (X∗, ‖ · ‖∗) and anisomorphism (which is an isometry) h : X → X∗ such that h(X) is dense in X∗.

2.4. Lebesgue spaces. In this section we shall discuss briefly Lebesgue spacesgenerated by spaces of continuous functions whose domain is RN , N ∈ N. If

f : RN → K, K = R or C

we define the support of f to be the closed set

supp(f) := x : f(x) 6= 0.

We say that f has compact support whenever supp(f) is a compact (i.e., closed andbounded) set and denote by C0(RN ) the set of all continuous K− valued functionsdefined on RN having compact support. (More generally, if Ω is an open set inRN , one denotes by Cj0(Ω) the set of all Cj− functions having compact supportin Ω.) This is a vector subspace of C(RN ), the space of all continuous K− valuedfunctions defined on RN .

We first need to define the Riemann integral on C0(RN ). To do this, withoutgetting into too many details of this procedure, we assume that the reader is familiarwith this concept for the integral defined on closed rectangular boxes

B := x = (ξ1, . . . , ξN ) : αi ≤ ξi ≤ βi, 1 ≤ i ≤ N,

(B =∏Ni=1[αi, βi]), where the numbers αi, βi, 1 ≤ i ≤ N , are fixed real numbers

(for each box). We observe that if f ∈ C0(RN ) and if B1 and B2 are such boxes,each of which contains supp(f), then∫

B1

f =∫B1∩B2

f =∫B2

f,

B1∩B2 also being a box containing supp(f). This allows us to define the Riemannintegral of f over RN by ∫

f(

=∫

RN

f)

:=∫B

f,

where B is any closed box containing supp(f).The mapping f 7→

∫f is a linear mapping (linear functional) from C0(RN ) to

K, which, in addition, satisfies

• if f is non-negative on RN , then∫f ≥ 0,


• If fn is a sequence of non-negative functions in C0(RN ) which is mono-tonically decreasing (pointwise) to zero, i.e.,

fn(x) ≥ fn+1(x), n = 1, . . . , limn→∞

fn(x) = 0, x ∈ RN ,

then∫fn → 0.

Definition 2.11. For f ∈ C0(RN ) we define

‖f‖1 :=∫|f |.

It is easily verified that ‖ · ‖1 is a norm - called the L1-normon C0(RN ). Wenow sketch the process for completing the normed vector space

(C0(RN ), ‖ · ‖1

)in

such a way that we may regard the vectors in the completion as functions on RN .

Definition 2.12. A subset S ⊂ RN is called a set of measure zero provided thatfor any ε > 0 there exists a sequence of boxes Bn∞n=1 such that

S ⊂ ∪∞n=1Bn,∞∑n=1

vol(Bn) < ε,

where vol(B) =∏Ni=1(βi − αi) for the box B =

∏Ni=1[αi, βi].

We say that a property holds “almost everywhere” (“a.e.”) if the set of pointsat which it fails to hold has measure zero.

The proofs of the following theorems may be found in a very complete discussionof C0(RN ) and its L1− completion in [37, Chapter 7].

Definition 2.13. A sequence xn in a normed vector space (X, ‖ · ‖) is said to bea fast Cauchy sequence if

∞∑n=1

‖xn+1 − xn‖

converges.

Theorem 2.14. If fn is a fast Cauchy sequence in (C0(RN ), ‖ · ‖1), then fnconverges pointwise a.e. in RN .

Definition 2.15. A Lebesgue integrable function on RN is a function f such that:• f is a K valued function defined a.e. on RN ,• there is a fast Cauchy sequence in (C0(RN ), ‖ ·‖1) which converges to f a.e.

in RN .

Theorem 2.16. If f is a Lebesgue integrable function and if fn and gn arefast Cauchy sequences in (C0(RN ), ‖ · ‖1) converging a.e. to f , then

limn→∞

∫fn = lim

n→∞

∫gn.

In light of this result we may define∫f by∫

f := limn→∞

∫fn,


where fn is any fast Cauchy sequence in (C0(RN ), ‖ · ‖1), converging a.e. to f onRN . The resulting map

f 7→∫f

is then defined on the space of all Lebesgue integrable functions L1(RN ) and is alinear functional on this space which also satisfies

|∫f | ≤

∫|f |, ∀f ∈ L1(RN ).

Theorem 2.17. The mapping

f 7→ ‖f‖1 :=∫|f |, f ∈ L1(RN ),

is a seminorm on L1(RN ), i.e., it satisfies all the conditions of a norm, except that‖f‖1 = 0 need not imply that f is the zero of L1(RN ). Further L1(RN ) is completewith respect to this seminorm and C0(RN ) is a dense subspace of L1(RN ).

Usually we identify two elements of L1(RN ) which agree a.e.; i.e., we define anequivalence relation on L1(RN )

f ∼ g

whenever the set A ∪B has measure zero, where

A := x : f(x) or g(x) fail to be defined,B := x : f(x), g(x) are defined, but f(x) 6= g(x).

This equivalence relation respects the operations of addition and scalar multiplica-tion and two equivalent functions have the same seminorm. The vector space of allequivalence classes then becomes a complete normed linear space (Banach space).This space, we again call L1(RN ),

Remark 2.18. 1. We again refer the reader to [37] for a complete discussion of thistopic and others related to it, e.g., convergence theorems for Lebesgue integrals,etc.

2. The ideas above may equally well be employed to define integrals on openregions Ω ⊂ RN starting with

C0(Ω) := f ∈ C(Ω) : supp(f) is a compact subset of Ω.The resulting space being L1(Ω).

3. One also may imitate this procedure to obtain the other Lebesgue spacesLp(RN ), 1 ≤ p <∞, by replacing the original norm in C0(RN ) by

‖f‖p :=( ∫

|f |p)1/p

, f ∈ C0(RN ).

And, of course, in similar vein, one can define Lp(Ω), 1 ≤ p <∞.4. For given f ∈ L1(R) define the functional Tf on C∞0 (R) as follows

Tf (φ) :=∫fφ.

The functional Tf is called the distribution defined by f . More generally, the set ofall linear functionals on C∞0 (R) is called the set of distributions on R and if T issuch, its distributional derivative ∂T is defined by

∂T (φ) := −T (φ′), ∀φ ∈ C∞0 (R),


hence for f ∈ L1(R) and Tf , the distribution determined by f ,

∂Tf (φ) =∫fφ′, ∀φ ∈ C∞0 (R).

We henceforth, for given f ∈ L1(R), we denote by f the distribution Tf determinedby f , as well.

5. The Cartesian product

E :=2∏i=1

L1(R)

may be viewed as a normed linear space with norm defined as

‖(u1, u2)‖ :=2∑i=1

‖ui‖1 ∀ui ∈ L1(Ω), i = 1, 2,

and the space C10 (R) may be viewed as a subspace of E by identifying f ∈ C1

0 (R)with (f, f ′). The completion of the latter space in E is called the Sobolev spaceW 1,1(R). Where we think of W 1,1(R) as a space of tuples of L1 functions. Onthe other hand, if F = (f, g) is such an element, then there exists a sequencefn ⊂ C1

0 (R) such that

fn → f, f ′n → g,

with respect to the L1 norm. It follows that∫fnφ→

∫fφ, ∀φ ∈ C∞0 (R),∫

f ′nφ→∫gφ, ∀φ ∈ C∞0 (R).

On the other hand, using integration by parts,∫f ′nφ = −

∫fnφ

′ → −∫fφ′

and therefore

−∫fφ′ =

∫gφ, ∀φ ∈ C∞0 (R).

I.e., in the sense of distributions ∂f = g. This may be summarized as follows: Thespace W 1,1(R) is the set of all L1 functions whose distributional derivatives are L1

functions, as well.If, instead of the L1 norm, we use the L2 norm in the above process, one obtains

the space W 1,2(R) which is usually denoted by H1(R). Using Lp as an underlyingspace, one may define the Sobolev spaces W 1,p(R), as well. In the case of functionsof N variables and open regions Ω ⊂ RN , analogous procedures are used to definethe Sobolev spaces W 1,p(Ω). Of particular interest to us later in these notes will bethe space H1

0 (Ω) which is the closure in H1(Ω) of the space C∞0 (Ω). We refer theinterested reader to the book by Adams [1] for detailed developments and propertiesof Sobolev spaces.


2.5. The Hausdorff metric. Let M be a metric space with metric d. For x ∈ Mand δ > 0, we set, as before,

B(x, δ) := y ∈ M : d(x, y) < δ,B[x, δ] := y ∈ M : d(x, y) ≤ δ,

the open and closed balls with center at x and radius δ. As pointed out before, theclosed ball is closed, but need not be the closure of the open ball.

Let A be a nonempty closed subset of M. For δ > 0 we define

Aδ : = ∪B[y, δ] : y ∈ A= x ∈ M : d(x, y) ≤ δ, for some y ∈ A.

We observe thatAδ ⊂ x ∈ M : d(x,A) ≤ δ,

whered(x,A) := infd(x, a) : a ∈ A.

If A is compact these sets are equal; if A is not compact, the containment may beproper.

Definition 2.19. We let

H := H(M) = A ⊂ M : A 6= ∅, A is closed and bounded .For each pair of sets A,B in H we define

D1(A,B) := supd(a,B) : a ∈ A, (2.3)

D2(A,B) := infε > 0 : A ⊂ Bε, . (2.4)

It is a straightforward exercise to prove the following proposition.

Proposition 2.20. For A,B ∈ H, D1(A,B) = D2(A,B).

We henceforth denote the common value

D(A,B) := D1(A,B) = D2(A,B). (2.5)

Proposition 2.21. For A,B ∈ H let h : H×H → [0,∞) be defined by

h(A,B) := D(A,B) ∨D(B,A) := maxD(A,B),D(B,A). (2.6)

Then h is a metric on H (the Hausdorff metric).

We briefly sketch the proof.That h is symmetric with respect to its arguments and that

h(A,B) = 0

if, and only if, A = B, follow easily.To verify that the triangle inequality holds, we let A,B,C ∈ H and let

x ∈ A, y ∈ B, z ∈ C.Then

d(x, z) ≤ d(x, y) + d(y, z),and hence,

d(x,C) ≤ d(x, y) + d(y, z), ∀y ∈ B, ∀z ∈ C.Therefore,

d(x,C) ≤ d(x,B) + D(B,C),


which implies that

D(A,C) ≤ D(A,B) + D(B,C) ≤ h(A,B) + h(B,C),

and similarlyD(C,A) ≤ h(A,B) + h(B,C).

The following corollary, which is an immediate consequence of the definitions,will be of use later.

Proposition 2.22. Let A,B ∈ H, a ∈ A, and η > 0 be given. Then there existsb ∈ B such that

d(a, b) ≤ h(A,B) + η

The following examples will serve to illustrate the computation of the Hausdorffdistance between two closed sets.

Example 2.23. Let

A := [0, 1]× 0, B := 0 × [1, 2] ⊂ R2,

thenD(A,B) =

√2, D(B,A) = 2,

so h(A,B) = 2.

Example 2.24. Let

A := B[a, r], B := B[b, s], a, b ∈ M, 0 < r ≤ s,

thenh(A,B) = d+ s− r,

where d = d(a, b).

There is a natural mapping associating points of M with elements of H given by

x 7→ x.This mapping, as one easily verifies, is an isometry, i.e.,

d(x, y) = h(x, y), ∀x, y ∈ M.

We next establish that (H,h) is a complete metric space, whenever (M,d) isa complete metric space (see also [25], which contains many very good exercisesconcerning the Hausdorff metric and the hierarchy of metric spaces constructed inthe above manner).

Let An ⊂ H be a sequence of sets such that

h(An, An+1) < 2−n, n = 1, 2, . . . .

We call a sequence xn ⊂ M, xn ∈ An, n = 1, 2, . . . a fast convergent sequence,provided that

d(xn, xn+1) < 2−n, n = 1, 2, . . . .We have the following lemma whose proof follows immediately from the definitionof the Hausdorff metric.

Lemma 2.25. Let (M,d) be a complete metric space and An ⊂ H be a sequenceof sets such that h(An, An+1) < 2−n, n = 1, 2, . . . . If j is a given positive integerand y ∈ Aj, then there exists a fast convergent sequence xn ⊂ M, xn ∈ An, n =1, 2, . . . with xj = y.


To see the above one proceeds as follows: Let xi = y and induct backwards tox1 and then induct forward through xi+1, . . . .

Theorem 2.26. If (M,d) is a complete metric space, then (H,h) is a completemetric space.

Proof. Let An ⊂ H be a Cauchy sequence. Then, by passing to a subsequence,we may assume that

h(An, An+1) < 2−n, n = 1, 2, . . . .

LetA := x ∈ M : x = lim

i→∞xi,

where xi ⊂ M, xi ∈ Ai, i = 1, 2, . . . , is a fast convergent sequence. We claimthat the closure of A, A, is an element of H and

Ai → A.

with respect to the Hausdorff metric. To establish that A is an element of H, itsuffices to show that A is a bounded set. Thus let x, y ∈ A and let xi ⊂ M, xi ∈Ai, i = 1, 2, . . . , and yi ⊂ M, yi ∈ Ai, i = 1, 2, . . . be fast convergent sequenceswith

limi→∞

xi = x, limi→∞

yi = y.

Then

d(x, x1) = limn→∞

d(xn, x1) ≤ limn→∞

k=n−1∑k=1

d(xk, xk+1) < 1,

and similarly d(y, y1) < 1. Hence

d(x, y) ≤ d(x, x1) + d(y, y1) + d(x1, y1),

ord(x, y) ≤ 2 + supd(v, w) : v, w ∈ A1 <∞.

We next note that h(An, A) → 0, if, and only if,

D(An, A) → 0 and D(A,An) → 0,

which, in turn, is equivalent to

supy∈An

d(y,A) → 0 and supz∈A

d(A,An) → 0.

For given y ∈ An, there exists a fast convergent sequence xi, xn = y, xi ∈ Ai,with, say, xi → x ∈ A. Hence,

d(y,A) ≤ d(y, x) = d(xn, x) ≤ 2−n+1,

so supy∈And(y,A) → 0, as n→∞.

Let z ∈ A. Then there exists a fast convergent sequence xi, xi ∈ Ai, with,say xi → z. Thus, for each n = 1, 2, . . . ,

d(z,An) ≤ d(z, xn) ≤ 2−n+1,

and consequently supz∈A d(z,An) → 0 as n→∞.

As a consequence of this result we also obtain the following theorem.

Theorem 2.27. If (M,d) is a metric space which is compact, then (H,h) is acompact metric space.


Proof. Since M is compact it is complete (see, e.g., [62]). It follows from the previoustheorem that H is complete. Thus we need to show that H is totally bounded.

Fix ε > 0, and choose 0 < δ < ε. Since M is compact, there exists a finite subsetS ⊂ M such that

M = ∪B(x, δ) : x ∈ S.If we denote by S := 2S\∅, the set of nonempty subsets of S, then S is a finite setand one can easily show that

H = ∪B(A, ε) : A ∈ S,where B(A, ε) is the ball, centered at A ∈ H with Hausdorff metric radius ε. HenceH is totally bounded and, since complete, also compact.

3. Contraction mappings

Let (M,d) be a complete metric space and let

T : M → Mbe a mapping. We call T a Lipschitz mapping with Lipschitz constant k ≥ 0,provided that

d(T (x), T (y)) ≤ kd(x, y), ∀x, y ∈ M. (3.1)We note that Lipschitz mappings are necessarily continuous mappings and that theproduct of two Lipschitz mappings (defined by composition of mappings) is againa Lipschitz mapping. Thus for a Lipschitz mapping T , and for all positive integersn, the mapping Tn = T · · · T , the mapping T composed with itself n times,is a Lipschitz mapping, as well. We call a Lipschitz mapping T a nonexpansivemapping provided the constant k may be chosen so that k ≤ 1, and a contractionmapping provided the Lipschitz constant k may be chosen so that 0 ≤ k < 1. Inthis case the Lipschitz constant k is also called the contraction constant of T .

3.1. The contraction mapping principle. In this section we shall discuss thecontraction mapping principle or what is often also called the Banach fixed pointtheorem. We shall also give some extensions and examples.

We have the following theorem.

Theorem 3.1. Let (M,d) be a complete metric space and let T : M → M be acontraction mapping with contraction constant k. Then T has a unique fixed pointx ∈ M. Furthermore, if y ∈ M is arbitrarily chosen, then the iterates xn∞n=0,given by

x0 = y

xn = T (xn−1), n ≥ 1,

have the property that limn→∞ xn = x.

Proof. Let y ∈ M be an arbitrary point of M and consider the sequence xn∞n=0,given by

x0 = y

xn = T (xn−1), n ≥ 1.

We shall prove that xn∞n=0 is a Cauchy sequence in M. For m < n we use thetriangle inequality and note that

d(xm, xn) ≤ d(xm, xm+1) + d(xm+1, xm+2) + · · ·+ d(xn−1, xn).


Since T is a contraction mapping, we have that

d(xp, xp+1) = d(T (xp−1), T (xp)) ≤ kd(xp−1, xp),

for any integer p ≥ 1. Using this inequality repeatedly, we obtain

d(xp, xp+1) ≤ kpd(x0, x1);

hence,d(xm, xn) ≤

(km + km+1 + · · ·+ kn−1

)d(x0, x1),

i.e.,

d(xm, xn) ≤km

1− kd(x0, x1),

whenever m ≤ n. From this we deduce that xn∞n=0 is a Cauchy sequence in M.Since M is complete, this sequence has a limit, say x ∈ M. On the other hand,since T is continuous, it follows that

x = limn→∞

xn = limn→∞

T (xn−1) = T ( limn→∞

xn−1) = T (x),

and, thus, x is a fixed point of T .If x and z are both fixed points of T , we get

d(x, z) = d(T (x), T (z)) ≤ kd(x, z).

Since k < 1, we must have that x = z.The following is an alternate proof. It follows (by induction) that for any x ∈ M

and any natural number m

d(Tm+1(x), Tm(x)) ≤ kmd(T (x), x).

Letα := inf

x∈Md(T (x), x).

Then, if α > 0, there exists x ∈ M such that

d(T (x), x) <32α

and hence for any m

d(Tm+1(x), Tm(x)) ≤ km32α.

On the other hand,

α ≤ d(T (Tm(x)), Tm(x)) = d(Tm+1(x), Tm(x))

and thus, for any m ≥ 1,

α ≤ km32α

which is impossible, since k < 1. Thus α = 0.We choose a sequence xn (a minimizing sequence) such that

limn→∞

d(T (xn), xn)) = α = 0.

For any m,n the triangle inequality implies that

d(xn, xm) ≤ d(T (xn), xn)) + d(T (xm), xm)) + d(T (xn), T (xm)).

And hence(1− k)d(xn, xm) ≤ d(T (xn), xn)) + d(T (xm), xm)).


Which implies that xn is a Cauchy sequence and hence has a limit x in M. Onenow concludes that

d(T (x), x) = 0

and thus x is a fixed point of T .

It may be the case that T : M → M is not a contraction on the whole space M,but rather a contraction on some neighborhood of a given point. In this case wehave the following result:

Theorem 3.2. Let (M,d) be a complete metric space and let

B = x ∈ M : d(z, x) < ε,

where z ∈ M and ε > 0 is a positive number and let T : B → M be a mapping suchthat

d(T (y), T (x)) ≤ kd(x, y), ∀ x, y ∈ B,with contraction constant k < 1. Furthermore assume that

d(z, T (z)) < ε(1− k).

Then T has a unique fixed point x ∈ B.

Proof. While the hypotheses do not assume that T is defined on the closure B ofB, the uniform continuity of T allows us to extend T to a mapping defined on Bwhich is a contraction mapping having the same Lipschitz constant as the originalmapping. We also note that for x ∈ B,

d(z, T (x)) ≤ d(z, T (z)) + d(T (z), T (x)) < ε(1− k) + kε = ε,

and hence T : B → B. Hence, by Theorem 3.1, since B is a complete metric space,T has a unique fixed point in B which, by the above calculations, must, in fact, bein B.

3.2. Some extensions.

Example 3.3. Let us consider the space

M = x ∈ R : x ≥ 1

with metricd(x, y) = |x− y|, ∀x, y ∈ M,

and let T : M → M be given by

T (x) := x+1x.

Then, an easy computation shows that

d(T (x), T (y)) =xy − 1xy

|x− y| < |x− y| = d(x, y).

On the other hand, there does not exist 0 ≤ k < 1 such that

d(T (x), T (y)) ≤ kd(x, y), ∀x, y ∈ M,

and one may verify that T has no fixed points in M.


This shows that if we replace the assumption of the theorem that T be a con-traction mapping by the less restrictive hypothesis that

d(T (x), T (y)) < d(x, y), ∀x, y ∈ M,

then T need not have a fixed point. On the other hand, we have the following resultof Edelstein [27] (see also [28]):

Theorem 3.4. Let (M,d) be a metric space and let T : M → M be a mapping suchthat

d(T (x), T (y)) < d(x, y), ∀x, y ∈ M, x 6= y.

Furthermore assume that there exists z ∈ M such that the iterates xn∞n=0, givenby

x0 = z

xn = T (xn−1), n ≥ 1,

have the property that there exists a subsequence xnj∞j=0 of xn∞n=0, with

limj→∞

xnj = y ∈ M.

Then y is a fixed point of T and this fixed point is unique.

Proof. We note from the definition of the iteration process that we may write

xn = Tn(x0),

where, as before, Tn is the mapping T composed with itself n times. We abbreviateby

yj = Tnj (x0) = Tnj (z),where the sequence nj is given by the theorem. Let us assume T has no fixedpoints. Then the function f : M → R defined by

x 7→ d(T 2(x), T (x))d(T (x), x)

is a continuous function. Since the sequence yj∞j=1 converges to y, the set Kgiven by

K = yj∞j=1 ∪ yis compact and, hence, its image under f is compact.

On the other hand, since,

f(x)d(T (x), x) = d(T 2(x), T (x)) < d(T (x), x), ∀ x ∈ M,

it follows that f(x) < 1, ∀ x ∈ M and, since K is compact, there exists a positiveconstant k < 1 such that

f(x) ≤ k, ∀x ∈ K.We now observe that for any positive integer m we have that

d(Tm+1(z), Tm(z)) =(m−1∏i=0

f(T i(z)))d(T (z), z).

Hence, for m = nj , we have

d(T (Tnj (z)), Tnj (z)) =( nj−1∏

i=0

f(T i(z)))d(T (z), z),


and, since f(T i(z)) ≤ k < 1, we obtain that

d(T (yj), yj) ≤ kj−1d(T (z), z).

On the other hand, as j →∞, yj → y and by continuity

T (yj) → T (y),

and alsokj−1 → 0,

we obtain a contradiction to the assumption that T (y) 6= y.

The above result has the following important consequence.

Theorem 3.5. Let (M,d) be a metric space and let T : M → M be a mapping suchthat

d(T (x), T (y)) < d(x, y), ∀x, y ∈ M, x 6= y.

Further assume thatT : M → K,

where K is a compact subset of M. Then T has a unique fixed point in M.

Proof. Since K is compact, it follows that for every z ∈ M the sequence Tn(z)has a convergent subsequence. Hence Theorem 3.4 may be applied.

A direct way of seeing the above is the following. By hypothesis we have thatT : K → K, and the function

x 7→ d(T (x), x)

is a continuous function on K and must assume its minimum, say, at a point y ∈ K.If T (y) 6= y, then

d(T 2(y), T (y)) < d(T (y), y),

contradicting that d(T (y), y) is the minimum value. Thus T (y) = y.

In some applications it is the case that the mapping T is a Lipschitz mappingwhich is not necessarily a contraction, whereas some power of T is a contractionmapping (see e.g. the result of [75]). In this case we have the following theorem.

Theorem 3.6. Let (M,d) be a complete metric space and let T : M → M be amapping such that

d(Tm(x), Tm(y)) ≤ kd(x, y), ∀x, y ∈ M,

for some m ≥ 1, where 0 ≤ k < 1 is a constant. Then T has a unique fixed pointin M.

Proof. It follows from Theorem 3.1 that Tm has a unique fixed point z ∈ M. Thus

z = Tm(z)

implies thatT (z) = TTm(z) = Tm(T (z)).

Thus T (z) is a fixed point of Tm and hence by uniqueness of such fixed pointsz = T (z).


Example 3.7. Let the metric space M be given by

M = C[a, b],

the set of continuous real valued functions defined on the compact interval [a, b].This set is a Banach space with respect to the norm

‖u‖ = maxt∈[a,b]

|u(t)|, u ∈ M.

We define T : M → M by

T (u)(t) =∫ t

a

u(s)ds.

Then‖T (u)− T (v)‖ ≤ (b− a)‖u− v‖.

(Note that b− a is the best possible Lipschitz constant for T.) On the other hand,we compute

T 2(u)(t) =∫ t

a

∫ s

a

u(τ)dτds =∫ t

a

(t− s)u(s)ds

and, inductively,

Tm(u)(t) =1

(m− 1)!

∫ t

a

(t− s)m−1u(s)ds.

It hence follows that

‖Tm(u)− Tm(v)‖ ≤ (b− a)m

m!‖u− v‖.

It is therefore the case that Tm is a contraction mapping for all values of m forwhich

(b− a)m

m!< 1.

It, of course, follows that T has the unique fixed point u = 0.

3.3. Continuous dependence upon parameters. It is often the case in appli-cations that a contraction mapping depends upon other variables (parameters) also.If this dependence is continuous, then the fixed point will depend continuously uponthe parameters, as well. This is the content of the next result.

Theorem 3.8. Let (Λ, ρ) be a metric space and (M,d) a complete metric spaceand let

T : Λ×M → Mbe a family of contraction mappings with uniform contraction constant k, i.e.,

d (T (λ, x), T (λ, y)) ≤ kd(x, y), ∀λ ∈ Λ, ∀x, y ∈ M.

Further more assume that for each x ∈ M the mapping

λ 7→ T (λ, x)

is a continuous mapping from Λ to M. Then for each λ ∈ Λ, T (λ, ·) has a uniquefixed point x(λ) ∈ M, and the mapping

λ 7→ x(λ),

is a continuous mapping from Λ to M.


Proof. The contraction mapping principle may be applied for each λ ∈ Λ, thereforethe mapping λ 7→ x(λ), is well-defined. For λ1, λ2 ∈ Λ we have

d (x(λ1), x(λ2)) = d (T (λ1, x(λ1)), T (λ2, x(λ2)))

≤ d (T (λ1, x(λ1)), T (λ2, x(λ1))) + d (T (λ2, x(λ1)), T (λ2, x(λ2)))

≤ d (T (λ1, x(λ1)), T (λ2, x(λ1))) + kd (x(λ1)), x(λ2)) .

Therefore

(1− k)d (x(λ1), x(λ2)) ≤ d (T (λ1, x(λ1)), T (λ2, x(λ1))) .

The result thus follows from the continuity of T with respect to λ for each fixedx.

3.4. Monotone Lipschitz mappings. In this section we shall assume that M isa Banach space with norm ‖ · ‖, which also a Hilbert space, i.e, that M is an innerproduct space (over the field of complex numbers) (see [63], [66]) with inner product(·, ·), related to the norm by

‖u‖2 = (u, u), ∀u ∈ M.

We call a mapping T : M → M, a monotone mapping provided that

Re ((T (u)− T (v), u− v)) ≥ 0, ∀u, v ∈ M,

where Re(c) denotes the real part of a complex number c.The following theorem, due to Zarantonello (see [64]), gives the existence of

unique fixed points of mappings which are perturbations of the identity mappingby monotone Lipschitz mappings, without the assumption that they be contractionmappings.

Theorem 3.9. Let M be a Hilbert space and let

T : M → M,

be a monotone mapping such that for some constant β > 0

‖T (u)− T (v)‖ ≤ β‖u− v‖, ∀u, v ∈ M.

Then for any w ∈ M, the equation

u+ T (u) = w (3.2)

has a unique solution u = u(w), and the mapping w 7→ u(w) is continuous.

Proof. If β < 1, then the mapping

u 7→ w − T (u),

is a contraction mapping and the result follows from the contraction mapping prin-ciple. Next, consider the case that β ≥ 1. We note that for λ 6= 0, u is a solutionof

u = (1− λ)u− λT (u) + λw, (3.3)if, and only if, u solves (3.2). Let us denote by

Tλ(u) = (1− λ)u− λT (u) + λw.

It follows that

Tλ(u)− Tλ(v) = (1− λ)(u− v)− λ(T (u)− T (v)).


Using properties of the inner product, we obtain

‖Tλ(u)− Tλ(v)‖2 ≤ λ2β2‖u− v‖2 − 2Re (λ(1− λ)(T (u)− T (v), u− v))

+ (1− λ)2‖u− v‖2.

Therefore, if 0 < λ < 1, the monotonicity of T implies that

‖Tλ(u)− Tλ(v)‖2 ≤ (λ2β2 + (1− λ)2)‖u− v‖2.

Choosing

λ =1

β2 + 1,

We obtain that Tλ satisfies a Lipschitz condition with Lipschitz constant k givenby

k2 =β2

β2 + 1,

hence is a contraction mapping. The result thus follows by an application of thecontraction mapping principle. On the other hand, if u and v, respectively, solve(3.2) with right hand sides w1 and w2, then we may conclude that

‖u− v‖2 + 2Re ((T (u)− T (v), u− v)) + ‖T (u)− T (v)‖2 = ‖w1 − w2‖2.

The monotonicity of T , therefore implies that

‖u− v‖2 + ‖T (u)− T (v)‖2 ≤ ‖w1 − w2‖2,

from which the continuity of the mapping w 7→ u(w) follows.

3.5. Multivalued mappings. Let M be a metric space with metric d and let

T : M → H(M), (3.4)

which is a contraction with respect to the Hausdorff metric h, i.e.,

h(T (x), T (y)) ≤ kd(x, y), ∀x, y ∈ M, (3.5)

where 0 ≤ k < 1 is a constant. Such a mapping is called a contraction correspon-dence.

For such mappings we have the following extension of the contraction mappingprinciple. It is due to Nadler [55]. We note that the theorem is an existencetheorem, but that uniqueness of fixed points cannot be guaranteed (easy examplesare provided by constant mappings).

Theorem 3.10. Let T : M → H(M) with

h(T (x), T (y)) ≤ kd(x, y), ∀x, y ∈ M,

be a contraction correspondence. Then there exists x ∈ M such that x ∈ T (x).

Proof. The proof uses the Picard iteration scheme. Choose any point x0 ∈ M, andx1 ∈ T (x0). Then choose x2 ∈ T (x1) such that

d(x2, x1) ≤ h(T (x1), T (x0)) + k,

where k is the contraction constant of T (that this may be done follows from Propo-sition 2.22 of Chapter 2). We then construct inductively the sequence xn∞n=0 inM to satisfy

xn+1 ∈ T (xn), d(xn+1, xn) ≤ h(T (xn), T (xn−1)) + kn.


We then obtain, for n ≥ 1,

d(xn+1, xn) ≤ h(T (xn), T (xn−1)) + kn

≤ kd(xn, xn−1) + kn

≤ k(h(T (xn−1, T (xn−2)) + kn−1

)+ kn

≤ k2d(xn−1, xn−2) + 2kn

. . .

≤ knd(x1, x0) + nkn.

Using the triangle inequality for the metric d, we obtain, using the above compu-tation

d(xn+m, xn) ≤n+m−1∑i=n

d(xi+1, xi)

≤n+m−1∑i=n

(kid(x1, x0) + iki

)≤

( ∞∑i=n

ki)d(x1, x0) +

( ∞∑i=n

iki).

Since both∑∞i=0 k

i and∑∞i=0 ik

i converge, it follows that xn∞n=0 is a Cauchysequence in M, hence has a limit x ∈ M. We next recall the definition of theHausdorff metric (see Chapter 2, Section 2.5) and compute

d(xn+1, T (x)) ≤ h(T (xn), T (x)) ≤ kd(xn, x).

Since T (x) is a closed set and limn→∞ xn = x, it follows that

d(x, T (x)) = 0,

i.e., x ∈ T (x).

3.6. Converse to the theorem. In this last section of the chapter we discuss aresult of Bessaga [9] which provides a converse to the contraction mapping principle.We follow the treatment given in [41], see also [23] (this last reference is also a verygood reference to fixed point theory, in general, and to the topics of these notes, inparticular). We shall establish the following theorem.

Theorem 3.11. Let M 6= ∅ be a set, k ∈ (0, 1) and let

F : M → M.

Then:(1) If Fn has at most one fixed point for every n = 1, 2, . . . , there exists a

metric d such that

d(F (x), F (y)) ≤ kd(x, y), ∀x, y ∈ M.

(2) If, in addition, some Fn has a fixed point, then there is a metric d suchthat

d(F (x), F (y)) ≤ kd(x, y), ∀x, y ∈ Mand (M,d) is a complete metric space.

The proof of Theorem 3.11 will make use of the following lemma.


Lemma 3.12. Let F be a selfmap of M and k ∈ (0, 1). Then the following state-ments are equivalent:

(1) There exists a metric d which makes M a complete metric space such that

d(F (x), F (y)) ≤ kd(x, y), ∀x, y ∈ M.

(2) There exists a function φ : M → [0,∞) such that φ−1(0) is a singletonand

φ(F (x)) ≤ kφ(x), ∀x ∈ M. (3.6)

Proof. (1. ⇒ 2.) The contraction mapping principle implies that F has a uniquefixed point z ∈ M. Put

φ(x) := d(x, z), ∀x ∈ M.

(2.⇒ 1.) Define

d(x, y) := φ(x) + φ(y), x 6= y

d(x, x) := 0.

Then, one easily notes that d is a metric on M and that F is a contraction withcontraction constant k. Let xn ⊂ M be a Cauchy sequence. If this sequence hasonly finitely many distinct terms, it clearly converges. Hence, we may assume it tocontain infinitely many distinct terms. Then there exists a subsequence xnk

ofdistinct elements, and hence, since

d(xnk, xnm

) = φ(xnk) + φ(xnm

),it follows that

φ(xnk) → 0.

Since there exists z ∈ M such that φ(z) = 0, it follows that

d(xnk, z) → 0,

and therefore xn → z.

To give a proof of Theorem 3.11 it will therefore suffice to produce such a functionφ. To do this, we will rely on the use of the Hausdorff maximal principle (see [62]).

Let z ∈ M be a fixed point of Fn, as guaranteed by part 2. of the theorem.Uniqueness then implies that

z = F (z),as well.

For a given function φ defined on a subset of M we denote by Dφ its domain ofdefinition and we let

Φ := φ : Dφ → [0,∞) : z ∈ Dφ ⊂ M, φ−1(0) = z, F (Dφ) ⊂ Dφ.We note that, for the given z, if we put

Dφ∗ := z, φ∗(z) := 0,

then φ∗ ∈ Φ. Hence the collection is not empty. One next defines a partial orderon the set Φ as follows:

φ1 : φ2 ⇐⇒ Dφ1 ⊂ Dφ2 and φ2|Dφ1= φ1.

If Φ0 is a chain in (Φ,), then the set

D := ∪φ∈Φ0Dφ


is a set which is invariant under F , it contains z and if we define

ψ : D → [0,∞)

byψ(x) := φ(x), x ∈ Dφ,

then ψ is an upper bound for Φ0 with domain Dψ := D. Hence, by the Hausdorffmaximal principle, there exists a maximal element

φ0 : D0 := Dφ0 → [0,∞)

in (Φ,). We next show that D0 = M, hence completing the proof.This we proof indirectly. Thus, let x0 ∈ M \D0 and consider the set

O := Fn(x0) : n = 0, 1, 2, . . . .

If it is the case thatO ∩D0 = ∅,

then the elements Fn(x0) : n = 0, 1, 2, . . . must be distinct; for, otherwise z ∈ O.We define

Dφ := O ∪D0, φ|D0 := φ0, φ(Fn(x0)) := kn, n = 0, 1, 2, . . . .

Thenφ ∈ Φ, φ 6= φ0, φ0 φ,

contradicting the maximality of φ0. Hence

O ∩D0 6= ∅.

Let us setm := minn : Fn(x0) ∈ D0,

then Fm−1(x0) /∈ D0. Define

Dφ := Fm−1(x0) ∪D0.

ThenF (Dφ) = Fm(x0) ∪ F (D0) ⊂ D0 ⊂ Dφ.

So Dφ is F invariant and contains z. Define φ : Dφ → [0,∞) as follows:• φ|D0 : φ0.• If Fm(x0) = z, put φ(Fm−1(x0)) := 1.• If Fm(x0) 6= z, put φ(Fm−1(x0)) := φ0(F

m(x0))k .

With this definition we obtain again a contradiction to the maximality of φ0 andhence must conclude that D0 = M.

Part 2. Applications

4. Iterated function systems

In this chapter we shall discuss an application of the contraction mapping prin-ciple to the study of iterated function systems. The presentation follows the workof Hutchinson [40] who established that a finite number of contraction mappingson a complete metric space M define in a natural way a contraction mapping ona subspace of H(M) with respect to the Hausdorff metric (see Chapter 2, Section2.5).


4.1. Set-valued contractions. Let M be a complete metric space with metric dand let

C(M) ⊂ H(M)

be the metric space of nonempty compact subsets of M endowed with the Hausdorffmetric h. Then if An is a Cauchy sequence in C(M), its limit A belongs to H(M)and hence is a closed and thus complete set. On the other hand, since An → A,for given ε > 0, there exists N , such that for n ≥ N , A ⊂ (An)ε, and hence A istotally bounded and therefore compact. Thus C(M) is a closed subspace of H(M),hence a complete metric space in its own right.


Theorem 4.1. Let fi : M → M, i = 1, 2, . . . k, be k mappings which are Lipschitzcontinuous with Lipschitz constants L1, L2, . . . , Lk, i.e.,

d (fi(x), fi(y)) ≤ Lid(x, y), i = 1, 2, . . . , k, x, y ∈ M. (4.1)

Define

F : C(M) → C(M) (4.2)

by

F (A) := ∪ki=1fi(A), A ∈ C(M). (4.3)

Then F satisfies a Lipschitz condition, with respect to the Hausdorff metric, withLipschitz constant

L := maxi=1,2,...,k

Li,

i.e.,

h (F (A), F (B)) ≤ Lh(A,B), ∀A,B ∈ C(M). (4.4)

In particular, if fi, i = 1, 2, . . . , k, are contraction mappings on M, then F , givenby (4.3), is a contraction mapping on C(M) with respect to the Hausdorff metric,and F has a unique fixed point A ∈ C(M).

Proof. We present two arguments based on the two equivalent definitions of theHausdorff metric. In both cases we establish the result for the case of two mappings.The general case will follow using an induction argument.

We first observe that for any A ∈ C(M), because of the compactness of A andthe Lipschitz continuity of fi, i = 1, 2 it is the case that fi(A) ∈ C(M), i = 1, 2and hence F (A) ∈ C(M).

Let us recall the definition of the Hausdorff metric which may equivalently bestated as

h(A,B) = supd(a,B),d(b, A), a ∈ A, b ∈ B.

Suppose then that A1, A2, B1, B2 are compact subsets of M. Letting A = A1 ∪A2, B = B1 ∪B2, we claim that

h(A,B) ≤ maxi=1,2

h(Ai, Bi) =: m. (4.5)

To see this, we let a ∈ A, b ∈ B and show that

d(a,B),d(b, A) ≤ m.


Now

d(a,B) = d(a,B1 ∪B2) = mind(a,Bi), i = 1, 2≤ d(a,Bi) ≤ D(Ai, Bi)

≤ h(Ai, Bi) ≤ m.

and similarly d(b, A) ≤ m, which establishes the claim.Let A,B ∈ C(M) and let h(A,B) = ε. Then

A ⊂ Bε, B ⊂ Aε,

where Aε, Bε are defined at the beginning of Section 2.5 of Chapter 2. We thereforehave

f1(B) ⊂ f1(Aε), f2(B) ⊂ f2(Aε)

f1(A) ⊂ f1(Bε), f2(A) ⊂ f2(Bε).(4.6)

It also follows thatfi(Aε) ⊂ (fi(A))Liε

, i = 1, 2

fi(Bε) ⊂ (fi(B))Liε, i = 1, 2.

(4.7)

Further, we obtain

fi(A) ⊂ (f1(B) ∪ f2(B))Lε , i = 1, 2

fi(B) ⊂ (f1(A) ∪ f2(A))Lε , i = 1, 2.(4.8)

Using again the definition of Hausdorff distance, it follows from (4.8) that

h (F (A), F (B)) ≤ Lε = Lh(A,B). (4.9)

An alternate argument is contained in the following. It follows from the discus-sion in Chapter 2 (see formula (2.6) there) that h (F (A), F (B)) is given by

h (F (A), F (B)) = D (F (A), F (B)) ∨D(F (B), F (A)) . (4.10)

Formula (4.5), on the other hand implies, since,

h (F (A), F (B)) = h (f1(A) ∪ f2(A), f1(B) ∪ f2(B)) , (4.11)

thath (F (A), F (B)) ≤ h (f1(A), f1(B)) ∨ h (f2(A), f2(B)) . (4.12)

Then, using the definition of the Hausdorff metric, we find that

h (fi(A), fi(B)) ≤ Lih(A,B), i = 1, 2. (4.13)

Combining (4.13) with (4.12), we obtain (4.4) for k = 2.

Remark 4.2. Given the contraction mappings f1, f2, . . . , fk, the mapping F , de-fined by

F (A) := ∪ki=1fi(A), A ∈ C(M)

has become known as the Hutchinson operator and the iteration scheme

Ai+1 = F (Ai), i = 0, 1, . . . (4.14)

an iterated function system. The iteration scheme (4.14), of course, has, by thecontraction mapping principle, a unique limit A, which is independent of the choiceof the initial set A0 and satisfies

A = F (A) = f1(A) ∪ f2(A) ∪ · · · ∪ fk(A). (4.15)


If it is the case that that M is a compact subset of RN and f1, f2, . . . , fk aresimilarity transformations, formula (4.15) says that the fixed set A is the union ofk similar copies of itself.

4.2. Examples. In this section we shall gather some examples which will illustratethe utility of the above theorem in the study and use of fractals.

The Cantor set. Let M = [0, 1] ⊂ R, with the metric given by the absolute value.We define

F : H(M) → H(M) = C(M)

byF (A) = f1(A) ∪ f2(A),

where

f1(x) =13x, f2(x) =

13x+

23, 0 ≤ x ≤ 1.

Then f1 and f2 are contraction mappings with the same contraction constant 13 .

Hence, F has the same contraction constant, also.The unique fixed point of F must satisfy

A = F (A) = f1(A) ∪ f2(A).

Considering the nature of the two transformations (similarity transformations), onededuces from the last equation, that the fixed point set A must be the Cantor subsetof [0, 1].

It is apparent how other types of Cantor subsets of an interval may be constructedusing other types of linear contraction mappings.

The Sierpinski triangle. Let M = [0, 1] × [0, 1] ⊂ R, with metric given by theEuclidean distance. We define

F : H(M) → H(M) = C(M)

byF (A) = f1(A) ∪ f2(A) ∪ f3(A),

where

f1(x, y) =12(x, y),

f2(x, y) =12(x, y) +

(12, 0

),

f3(x, y) =12(x, y) +

(14,12).

Here, again, all three contraction constants are equal to 12 , hence, the Hutchinson

operator is a contraction mapping with the same contraction constant. All threemappings are similarity transformations and, since the fixed point A of F satisfies

A = F (A) = f1(A) ∪ f2(A) ∪ f3(A),

A equals a union of three similar copies of itself, i.e., it is a self-similar set, whichin this case is, what has become known the Sierpinski triangle.

For many detailed examples of the above character, we refer to [6], [57].


5. Newton’s method

One of the important numerical methods for computing solutions of nonlin-ear equations is Newton’s method, often also referred to as the Newton-Raphsonmethod. It is an iteration scheme, whose convergence may easily be demonstratedby means of the contraction mapping principle. Many other numerical methodscontain Newton’s method as one of their subroutines (see, e.g., [2]).

Let G be a domain in RN and let

F : G→ RN

be C2 mapping (i.e., all first and second partial derivatives of all components of Fare continuous on G).

Let us assume that the equation

F (x) = 0, (5.1)

has a solution x∗ ∈ G such that the Jacobian matrix F ′(x∗) has full rank (i.e.,the matrix F ′(x∗) is a nonsingular matrix). It then follows by a simple continuityargument that F ′(x) has full rank in a closed neighborhood of x∗, say

Br := x : ‖x∗ − x‖ ≤ r, r > 0,where ‖ · ‖, is a given norm in RN , and that x∗ is the unique solution of (5.1) there.The mapping

x 7→ x− (F ′(x))−1F (x) =: N(x) (5.2)

is therefore defined in that neighborhood and we note that x∗ is a solution of (5.1)in that neighborhood if, and only if, x∗ is a fixed point of N in Br.

The Newton iteration scheme is then defined by:

xn+1 = N(xn), x1 ∈ Br, n = 1, 2, . . . . (5.3)

The following theorem holds.

Theorem 5.1. Assume the above conditions hold. Then, for all r > 0, sufficientlysmall, the Newton iteration scheme, given by (5.3), converges to the solution x∗ of(5.1).

Proof. We use Taylor’s theorem to write

N(x) = N(x∗) +N ′(x∗)(x− x∗) +O(‖x− x∗‖2).On the other hand, because F is a C2 mapping and F ′(x∗) is nonsingular, we obtainthat

N(x∗ + y) = N(x∗) + y − (F ′(x∗ + y))−1F (x∗ + y)

= N(x∗) + y − (F ′(x∗))−1F (x∗ + y)

+((F ′(x∗))−1 − (F ′(x∗ + y))−1

)F (x∗ + y)

= N(x∗) +O(‖y‖2) + (F ′(x∗ + y))−1(F ′(x∗ + y)

− F ′(x∗))

(F ′(x∗))−1 (F ′(x∗)y +O(‖y‖2)

)= N(x∗) +O(‖y‖2).

From which it follows thatN ′(x∗) = 0,


the zero matrix, and thus, there exists r > 0, such that the matrix norm of N ′(x)satisfies

‖N ′(x)‖ ≤ 12, for ‖x− x∗‖ ≤ r.

Hence, for x, y ∈ Br

‖N(x)−N(y)‖ ≤∫ 1

0

‖N ′((1− t)y + tx)‖dt‖y − x‖ ≤ 12‖x− y‖

and for y = x∗

‖x∗ −N(x)‖ ≤∫ 1

0

‖N ′((1− t)x∗ + tx)‖dt‖x∗ − x‖ ≤ r.

Hence, N : Br → Br and N is a contraction mapping for such r. The assertion ofthe theorem then follows from the contraction mapping principle.

6. Hilbert’s metric

This chapter is concerned with a fundamental result of matrix theory, the the-orem of Perron-Frobenius about the existence of positive eigenvectors of positivematrices. Upon the introduction of Hilbert’s metric, the result may be deducedvia the contraction mapping principle. Since the approach also works in infinitedimensions, a version of the celebrated theorem of Krein-Rutman may be estab-lished, as well. The approach to establishing these important results using Hilbert’sprojective metric goes back to Birkhoff [10]. Here we also rely on the work in [16]and [45].

6.1. Cones. Let E be a real Banach space with norm ‖ · ‖. A closed subset K inE is called a cone provided that:

(1) for all λ, µ ≥ 0, and all u, v ∈ K λu+ µv ∈ K,(2) if u ∈ K, −u ∈ K, then u = 0 ∈ K.

If it is the case that the interior of K, intK, is not empty, the cone is calledsolid. In this chapter we shall always assume that the cone K is solid, even thoughseveral of the results presented are valid in the absence of this assumption.

A cone K induces a partial order ≤ by:

u ≤ v if, and only if, v − u ∈ K,and if the cone K is solid, another partial order < by:

u < v if, and only if v − u ∈ intK.

Since we assume that a cone is closed, it follows that it is also Archimedean, i.e.,

if nu ≤ v, n = 1, 2, . . . , then u ≤ 0.

We shall denote by K+ the set of all nonzero elements of K.For solid cones we have the following lemma (see [65] for most of the details).

Lemma 6.1. Let K be a solid cone and let v ∈ intK, u ∈ K. Then:(1) w : w = (1− t)v + tu, 0 ≤ t < 1 ⊂ intK.(2) If u ∈ ∂K, then for t > 1,

w = (1− t)v + tu 6∈ K.(3)

K = intK.


(4)

K + v ⊂ intK.

Proof. We shall establish the last part of the lemma and leave the remaining partsto the reader for verification.

If v ∈ intK and z ∈ K, then v + z = u ∈ K. If it were the case that u ∈ ∂K,then by the first two assertions of the lemma t = 1 is the maximal number suchthat

w = (1− t)v + tu ∈ K.

On the other hand

w = (1− t)v + tu = (1− t)v + t(v + z) = v + tz ∈ K, ∀t ≥ 0,

yielding a contradiction.

6.2. Hilbert’s metric. We define the mappings

m(·, ·), M(·, ·), [·, ·] : E ×K+ → [−∞,∞]

as follows:

m(u, v) := supλ : λv ≤ u (6.1)

M(u, v) := infλ : u ≤ λv, (6.2)

with the interpretation that m(u, v) = −∞, if the set λ : λv ≤ u is empty, andM(u, v) = ∞, if the set λ : u ≤ λv is empty, and

[u, v] := M(u, v)−m(u, v). (6.3)

The last quantity is called the v−oscillation of u. We remark that the Archim-edean property immediately implies that

m(u, v) <∞, M(u, v) > −∞,

which makes (6.3) well-defined in the extended real numbers.In what is to follow many of the statements are to be interpreted in the extended

real numbers. If this should be the case, we shall not remark so explicitly; it willbe clear from the context. The following lemma is easy to prove and we leave thedetails to the reader. We remark that (6.6) follows from Lemma 6.1 (above).


Lemma 6.2. For u, v, w ∈ K+, the following hold:

m(u, v)v ≤ u ≤M(u, v)v, provided M(u, v) <∞, (6.4)

0 ≤ m(u, v) ≤M(u, v) ≤ ∞, (6.5)

m(u, v) > 0, if u ∈ intK, and u−m(u, v)v ∈ ∂K, (6.6)

m(u, v) = 0, if v ∈ intK, and u ∈ ∂K, (6.7)

M(u, v) <∞, if v ∈ intK, (6.8)

M(u,w) ≤M(u, v)M(v, w), (6.9)

m(u,w) ≥ m(u, v)m(v, w), (6.10)

m(u, v)M(v, u) = 1, (6.11)

M(λu+ µv, v) = λM(u, v) + µ, ∀λ, µ ≥ 0, (6.12)

m(λu+ µv, v) = λm(u, v) + µ, ∀λ, µ ≥ 0, (6.13)

[λu+ µv, v] = λ[u, v], ∀λ, µ ≥ 0. (6.14)

[u, v] = 0, implies u = λv, for some λ ≥ 0. (6.15)

Using the properties in the previous lemma, one may establish the followingresult.

Lemma 6.3. For u, v ∈ K+, the following hold:

M(u, u+ v) =1

m(u+ v, u)

=1

1 +m(v, u)

=M(u, v)

1 +M(u, v)≤ 1,

(6.16)

and

m(u, u+ v) =1

M(u+ v, u)

=1

1 +M(v, u)

=m(u, v)

1 +m(u, v)≤ 1.

(6.17)

Hence

M(u, u+ v) +m(v, u+ v) = 1. (6.18)

We now define Hilbert’s projective metric

d : intK × intK → [0,∞)

as follows:

d(u, v) := logM(u, v)m(u, v)

. (6.19)



Theorem 6.4. The function d defined by (6.19) has the following properties: Ifu, v, w ∈ intK, then

d(u, v) = d(v, u), d(λu, µv) = d(u, v), ∀λ > 0, µ > 0, (6.20)

d(u, v) = 0, if, and only if, u = λv, for some λ > 0, (6.21)

d(u, v) ≤ d(u,w) + d(w, v). (6.22)

LetM := u ∈ intK : ‖u‖ = 1, (6.23)

then (M,d) is a metric space.

Proof. The symmetry property (6.20) and the triangle inequality (6.22) follow im-mediately from Lemma 6.3. That (6.21) holds follows from the fact that d(u, v) = 0,if, and only if, m(u, v) = M(u, v), which is the case, if, and only if, u = M(u, v)v.The properties together imply that d is a metric on M.

The following examples will serve to illustrate these concepts. In all exampleswe shall assume that u, v ∈ intK.

Example 6.5. Let

E := RN , K := (u1, u2, . . . , uN ) : ui ≥ 0, i = 1, 2, . . . , N.Then

intK = (u1, u2, . . . , uN ) : ui > 0, i = 1, 2, . . . , N,

m(u, v) = mini

uivi, M(u, v) = max

i

uivi,

d(u, v) = log maxi,j

uivjujvi

.

Example 6.6. Let

E := RN , K := (u1, u2, . . . , uN ) : 0 ≤ u1 ≤ u2 ≤ · · · ≤ uN.Then

intK = (u1, u2, . . . , uN ) : 0 < u1 < u2 < · · · < uN,

m(u, v) = mini<j

uj − uivj − vi

, M(u, v) = maxi<j

uj − uivj − vi

,

d(u, v) = log maxi<j,k<l

(uj − ui)(vl − vk)(ul − uk)(vj − vi)

.

Example 6.7. Let

E := C[0, 1], K := u ∈ E : u(x) ≥ 0, 0 ≤ x ≤ 1.Then

intK = u ∈ E : u(x) > 0, 0 ≤ x ≤ 1,

m(u, v) = minx∈[0,1]

u(x)v(x)

, M(u, v) = maxx∈[0,1]

u(x)v(x)

,

d(u, v) = log max(x,y)∈[0,1]2

u(x)v(y)u(y)v(x)

.


It is an instructive exercise to compute the various quantities m(u, v),M(u, v),etc., also in the cases that u, v are not necessarily interior elements to the cone K.

6.3. Positive mappings. A mapping T : E → E is called a positive mapping(with respect to the cone K) provided that

T (K+) ⊂ K+.

A positive mapping T is called homogeneous of degree p, p ≥ 0, whenever

T (λu) = λpT (u), ∀λ > 0, u ∈ K.A positive mapping is called monotone provided that

u, v ∈ K, u ≤ v, imply T (u) ≤ T (v).

In the following we are interested to see under what conditions positive mappingsare contractions with respect to Hilbert’s projective metric. In order to achieve this,we shall derive some properties of positive mappings with respect to the functionsintroduced above.

We have the following lemma.

Lemma 6.8. Let T be a positive monotone mapping which is homogeneous of degreep. Then for any u, v ∈ K+

m(u, v)p ≤ m(T (u), T (v)) ≤M(T (u), T (v)) ≤M(u, v)p, (6.24)

and ifk(T ) := infλ : d(T (u), T (v)) ≤ λd(u, v), d(u, v) <∞, (6.25)

where d is Hilbert’s projective metric, then

k(T ) ≤ p.

In particular:(1) If p < 1, then T is a contraction with respect to the projective metric.(2) If T is linear, then

d(T (u), T (v)) ≤ d(u, v), ∀u, v ∈ K+.

Proof. Sincem(u, v)v ≤ u ≤M(u, v)v,

Inequality (6.24) follows from the monotonicity and homogeneity of T . Using thedefinition of Hilbert’s projective metric and (6.24) we obtain that

d(T (u), T (v)) ≤ log(M(u, v)m(u, v)

)p= pd(u, v),

from which the result follows.

Remark 6.9. The constant k(T ), above, is called the contraction ratio of themapping T .

We next concentrate on computing the contraction ratio for positive linear map-pings T . We define the following constants.

∆(T ) := supd(T (u), T (v)) : u, v ∈ K+,

Γ(T ) :=e

12∆(T ) − 1e

12∆(T ) + 1

(6.26)


(∆(T ) is called the projective diameter of T ) and

N(T ) := infλ : [T (u), T (v)] ≤ λ[u, v], u, v ∈ K+. (6.27)

We next establish an extension of a result originally proved by Hopf ([38], [39])in his studies of integral equations and extended by Bauer [7] to the general setting.

Theorem 6.10. Let T : E → E be a linear mapping which is positive with respectto the cone K. Let u, v ∈ K+ be such that [u, v] <∞. Then

[T (u), T (v)] ≤ Γ(T )[u, v], (6.28)

where Γ(T ) is given by (6.23), i.e. N(T ) ≤ Γ(T ). Furthermore

k(T ) ≤ N(T ). (6.29)

Proof. If [u, v] = 0 (which is the case, if, and only if, u and v are co-linear), then[T (u), T (v)] = 0 and the result holds trivially.

In the contrary case, 0 < [u, v] <∞, and, since T is a positive operator, we havethat for any u, v ∈ K+, the images of the elements

p = u− (m(u, v))v,

q = (M(u, v))v − u,

T (p) and T (q) belong to K+. Then

p+ q = [u, v]v,

and (see Lemma 6.2)

m(T (u), T (v)) = [u, v]m(T (p), T (p) + T (q)) +m(u, v)

= νM(u, v) + (1− ν)m(u, v),(6.30)

where

ν = m(T (p), T (p) + T (q)).

Since T is a positive mapping, it follows from Lemma 6.3 that

ν =1

1 +M(T (q), T (p)).

We similarly obtain

M(T (u), T (v)) = µM(u, v) + (1− µ)m(u, v), (6.31)

where

µ = M(T (p), T (p) + T (q)) =1

1 +m(T (q), T (p)).

Hence

[T (u), T (v)] = (µ− ν)[u, v], (6.32)

where

µ− ν =M(T (p), T (q))M(T (q), T (p))− 1

(1 +M(T (p), T (q)))(1 +M(T (q), T (p)))=: φ(T (p), T (q)). (6.33)


We next observe that

φ(T (p), T (q) ≤ M(T (p), T (q))M(T (q), T (p))− 1(√M(T (p), T (q))M(T (q), T (p)) + 1

)2

=ed(T (p),T (q)) − 1(√ed(T (p),T (q)) + 1

)2

=

√ed(T (p),T (q)) − 1√ed(T (p),T (q)) + 1

≤ Γ(T ).

(6.34)

proving (6.28).To verify (6.29), we use the above together with the identities of Lemmas 6.2

and 6.3. Since[T (u), T (v)] ≤ N(T )[u, v],

we have1

m(T (v), T (u))− 1M(T (v), T (u))

≤ N(T )( 1m(v, u)

− 1M(v, u)

).

We now replace v by cv + u, c > 0 and use Lemma 6.2 to find

c[T (v), T (u)](cM(T (v), T (u)) + 1) (cm(T (v), T (u)) + 1)

≤ N(T )c[v, u]

(cM(v, u) + 1) (cm(v, u) + 1).

We integrate this inequality with respect to c and obtain

logcM(T (v), T (u)) + 1cm(T (v), T (u)) + 1

≤ N(T ) logcM(v, u) + 1cm(v, u) + 1

.

We let c→∞ and obtain

d(T (v), T (u)) ≤ N(T )d(v, u),

or, equivalentlyd(T (u), T (v)) ≤ N(T )d(u, v),

i.e., (6.29) holds, which completes the proof of the theorem.

We summarize the above results in the following theorem.

Theorem 6.11. Let T be a positive monotone mapping which is homogeneous ofdegree p. Then for any u, v ∈ K+, with d(u, v) <∞,

d(T (u), T (v)) ≤ pd(u, v), (6.35)

where d is Hilbert’s projective metric.In particular, if p < 1, then T is a contraction with respect to the projective

metric. If T is linear, then

d(T (u), T (v)) ≤ Γ(T )d(u, v), ∀u, v ∈ K+. (6.36)

Thus, in particular, if ∆(T ) <∞, where ∆(T ) and Γ(T ) are defined in (6.26), thenT is a contraction mapping with respect to Hilbert’s projective metric.


6.4. Completeness criteria. It follows from Theorem 6.11 that if T is a mapping,satisfying the hypotheses there, it will be a contraction mapping with respect tothe projective metric and, hence, if the mapping leaves the unit spheres of the coneK invariant and M is complete with respect to the topology defined by the metric,then the contraction mapping principle may be applied. We shall now describesituations where completeness prevails.

We shall discuss one such situation, namely, the case that the Banach space Eis a Banach space whose norm is monotone with respect to the cone K, i.e.,

u, v ∈ K, u ≤ v, then ‖u‖ ≤ ‖v‖

(e.g. if E is a Banach lattice, see [65], with respect to the partial order induced bythe cone K). Each of the cones in the examples discussed earlier generates such aBanach space, as do the cones of nonnegative functions in all Lp− spaces.

We have the following result.

Theorem 6.12. Let E be a real Banach space whose norm is monotone with respectto a solid cone K. Then

M := u ∈ intK : ‖u‖ = 1

is complete with respect to Hilbert’s projective metric d.

Proof. Assume that un is a Cauchy sequence in M with respect to the metric d,then for ε > 0, given, there exists an integer N , such that

n,m ≥ N, implies that 1 ≤ M(un, um)m(un, um)

≤ 1 + ε.

Furthermore, we have (see the definitions of m and M),

m(un, um)um ≤ un ≤M(un, um)um ≤ (1 + ε)m(un, um)um, (6.37)

and therefore (using the monotonicity of the norm),

11 + ε

≤ m(un, um) ≤ 1, n,m ≥ N.

We next use (6.37) to conclude that

0 ≤ un −m(un, um)um ≤ m(un, um)(ed(un,um) − 1

)um,

and, therefore‖un −m(un, um)um‖ ≤ (ed(un,um) − 1).

Thus

‖un − um‖ ≤ ‖un −m(un, um)um‖+ ‖m(un, um)um − um‖

≤ (ed(un,um) − 1) + (1−m(un, um))

≤ ε+ε

1 + ε,

(6.38)

proving that un is a Cauchy sequence in E. Since E is complete and the unitsphere of E and K are closed, this sequence will have a limit u ∈ K of norm 1. Wenext show that u ∈ intK. This follows from the fact that the boundary of K maybe characterized by

∂K = v ∈ K : m(v, w) = 0, ∀ w ∈ intK,


(see Lemma 6.2) and that for each u ∈ M, the mapping

v 7→ m(u, v)

M → [0, 1]

is an upper semicontinuous function with respect to the norm. To see this, itsuffices to show the sequential upper semicontinuity of m. Thus, let vn ⊂ M bea sequence with

vn → v,

and let u ∈ M. Letα = lim sup

n→∞m(u, vn),

then 0 ≤ α ≤ 1, and given ε ∈ (0, 1)

(1− ε)α ≤ m(u, vn) ≤ 1.

Hence,(1− ε)αvn ≤ u,

and consequently,(1− ε)αv ≤ u,

i.e.,(1− ε)α ≤ m(u, v),

showing that α ≤ m(u, v), proving the upper semicontinuity of m.Returning to the sequence un, above with un → u, we see that for fixed m,

m(u, um) ≥ lim supn→∞

m(un, um) ≥ 11 + ε

,

and, therefore, u ∈ intK. One may similarly verify that the mapping M , andhence, d are lower semicontinuous functions, which will further imply that

limn→∞

d(u, un) = 0.

6.5. Homogeneous operators. In this section we shall establish an eigenvaluetheorem for monotone positive operators which are homogeneous of degree lessthan one. We have.

Theorem 6.13. Let E be a real Banach space whose norm is monotone with respectto the cone K. Let

T : K+ → K+

be a monotone operator which is homogeneous of degree p < 1 and leaves the interiorof the cone, intK, invariant. Then for any positive number µ, there exists u ∈ intKsuch that

T (u) = µu. (6.39)

Proof. Let

f(u) :=T (u)‖T (u)‖

,

then f : M → M. We have, by the properties of Hilbert’s projective metric that

d(f(u), f(v)) ≤ pd(u, v), ∀u, v ∈ M.


Hence f is a contraction mapping. Since M is complete with respect to this metric,it follows from the contraction mapping theorem that f has a unique fixed point uin M, i.e.

u =T (u)‖T (u)‖

orT (u) = ‖T (u)‖u.

We let u = λy and obtain

T (y) = λ1−pry, r = ‖T (u)‖,

and for given µ choose λ such that µ = λ1−pr.

To provide an example illustrating the above result, we consider the following.Let

E := C[0, 1],

with the usual maximum norm, and let

G : [0, 1]2 → [0,∞)

be a nontrivial continuous function. Let T : E → E be given by

T (u)(t) :=∫ 1

0

G(t, s)|u(s)|p−1u(s)ds, (6.40)

where 0 < p < 1 is a constant. In E we may consider the solid cone

K := u ∈ E : u(t) ≥ 0, 0 ≤ t ≤ 1.

Then T is a monotone operator which is homogeneous of degree p and the norm ofE is monotone with respect to K. We hence have the following result.

Example 6.14. Let the above assumptions hold. Then for any µ ∈ (0,∞) thereexists a continuous function

u : [0, 1] → [0,∞)

with u : (0, 1) → (0,∞) solving the integral equation

µu(t) =∫ 1

0

G(t, s)|u(s)|p−1u(s)ds. (6.41)

6.6. On positive eigenvectors and eigenvalues. In this section we shall assumethat T : E → E is a linear operator which is positive with respect to the cone Kand satisfies

T (K+) ⊂ K+, T (intK) ⊂ intK. (6.42)

We shall establish the classical results of Perron-Frobenius and Krein-Rutman (see,for example [46]) about principal eigenvalues of a special class of such operators.

We call a positive linear operator S uniformly positive, provided there existsu0 ∈ intK and a constant β > 1 such that

λ(u)u0 ≤ S(u) ≤ βλ(u)u0, u ∈ intK, (6.43)

where λ(u) is a positive constant depending upon u.We have the following theorem.


Theorem 6.15. Let the norm of E be monotone with respect to the cone K andlet T be a linear positive operator satisfying (6.42) such that for some integer n,the operator Tn is uniformly positive. Then there exists a unique pair (µ, u) ∈(0,∞)×M such that

T (u) = µu. (6.44)

Proof. We define the mapping g : M → M by

g(u) :=T (u)‖T (u)‖

,

and letf := g · · · g︸︷︷︸

n

,

i.e., g composed with itself n times. Then

f(u) =S(u)‖S(u)‖

,

where S = Tn. It follows from the properties of Hilbert’s projective metric, that

d(f(u), f(v)) = d(S(u), S(v)), u, v ∈ M,

and that M is complete, since the norm of E is monotone with respect to the coneK. Thus, f will have a unique fixed point, once we show that f is a contractionmapping, which will follow from Theorem 6.11 once we establish that ∆(S) <∞.

To compute ∆(S), we recall the definition of projective diameter (see (6.26)) andfind that for any u, v ∈ M,

d(S(u), S(v)) ≤ d(S(u), u0) + d(S(v), u0)

and, therefore, by the uniform positivity of Tn,

d(S(u), u0), d(S(v), u0) ≤ log β,

implying thatd(S(u), S(v)) ≤ 2 log β.

Thus S, and hence, f , are contraction mappings with respect to the projectivemetric and therefore, there exists a unique u ∈ M such that

f(u) = u,

i.e. S(u) = u, orTn(u) = ‖Tn(u)‖u,

and the direction u is unique. Furthermore, since f has a unique fixed point inM, g will have a unique fixed point also, as follows from Theorem 3.6 of Chapter3. This also implies the uniqueness of the eigenvalue with corresponding uniqueeigenvector u ∈ intK, ‖u‖ = 1.

In the following we provide two examples to illustrate the above theorem. Thefirst example illustrates part of the Perron-Frobenius theorem and the second is anextension of this result to operators on spaces of continuous functions. We remarkhere that the second result concerns an integral equation which is not given by acompact linear operator (see also [10]).


Example 6.16. Let

E = RN , K = (u1, u2, . . . , uN ) : ui ≥ 0, i = 1, 2, . . . , N.

Let T : K → K be a linear transformation whose N ×N matrix representation isirreducible. Then there exists a unique pair (λ, u) ∈ (0,∞)× intK, ‖u‖ = 1, suchthat

Tu = λu.

Proof. An N × N matrix is irreducible (see [48]), provided there does not exist apermutation matrix P such that

PTPT =(B OC D

),

where B and D are square submatrices. This is equivalent to saying, that forsome positive integer n, the matrix Tn = (ti,j) has only positive entries ti,j , i, j =1, . . . , N . Since,

intK = (u1, u2, . . . , uN ) : ui > 0, i = 1, 2, . . . , N,

if we letm = min

i,jti,j , M = max

i,jti,j , u0 = (1, 1, . . . , 1),

then for any u ∈ K+,m‖u‖1u0 ≤ Tnu ≤M‖u‖1u0,

where

‖u‖1 =N∑i=1

|ui|,

is the l1 norm of the vector u. This shows that Tn is a uniformly positive operator.Hence Theorem 6.15 may be applied.

For many applications of positive matrices (particularly to economics) we referto [48], [70]. The following example is discussed in [10].

Let again E := C[0, 1], with the usual maximum norm and K the cone of non-negative functions. Let

p : [0, 1]2 → (0,∞)

be a continuous function. Let

0 < I := inf[0,1]2

p(x, y) ≤ sup[0,1]2

p(x, y) =: µI.

Supposeg : [0, 1] → [0, 1]

is a continuous function and define T : E → E by

T (u)(x) :=∫ 1

0

p(x, y)u(y)dy + au(g(x)), (6.45)

where a is a positive constant.We have the following example.


Example 6.17. Let T be defined by (6.45), where p, g, a satisfy the above condi-tions. Then there exists a unique positive number λ and a continuous function

u : [0, 1] → (0, 1], maxx∈[0,1]

u(x) = 1,

such that

λu(x) =∫ 1

0

p(x, y)u(y)dy + au(g(x)), 0 ≤ x ≤ 1. (6.46)

To see how the result of Example 6.17 follows from Theorem 6.15 we proceed asfollows.

We replace the cone K by the following subcone, which we denote by K1

K1 := u ∈ K : maxu ≤ νminu,

wheremaxu = max

x∈[0,1]u(x), minu = min

x∈[0,1]u(x),

and ν > µ.Easy computations show that the norm is monotone with respect to the new

cone K1, and that for any x ∈ [0, 1]

(I + 1)minu ≤ (Tu)(x) ≤ ν(µI + a) minu.

This inequality shows that the operator T is a uniformly positive operator, asrequired by the theorem. Furthermore, letting v = Tu, we obtain that

max v ≤ µI

∫ 1

0

udx+ aνminu,

min v ≥ I

∫ 1

0

udx+ aminu,

and thereforemax vmin v

≤µI

∫ 1

0udx+ aνminu

I∫ 1

0udx+ aminu

≤ ν,

showing that T : K1 → K1. We may, hence, apply Theorem 6.15. We remark thatthe operator T , above, is not a compact operator and hence techniques based onLeray-Schauder degree and fixed point theory may not be applied here.

Let us consider another situation, to which the results derived above apply.

Theorem 6.18. Let the norm of E be monotone with respect to the solid cone Kand let

T : K+ → intK,

be a positive, linear, and compact operator. Then T has a unique eigenvector in M.

Proof. Let v ∈ M be a fixed element. Then T (v) ∈ intK. Hence, there existpositive numbers ε > 0 and α, depending on v, such that

αv ≤ T (v), and B(v, ε) ⊂ intK.

For each integer n = 1, 2, . . . , we define the mapping Sn : M → M, by

Sn(u) :=T (u+ 1

nv)‖T (u+ 1

nv)‖.


It follows from earlier considerations, that Sn will be a contraction mapping withrespect to Hilbert’s metric, once we show that

d(Sn(u), v) ≤ c,

where c is a constant, independent of u ∈ M. To see this, we observe that

Sn(u) ≥T ( 1

nv)‖T (u+ 1

nv)‖≥ α

n(1 + 1n )‖T‖

v,

hence,m(Sn(u), v) ≥

α

n(1 + 1n )‖T‖

.

Furthermore, since v − εSn(u) ∈ intK, it follows that

M(Sn(u), v) ≤1ε.

Thus,

d(Sn(u), v) ≤ logn(1 + 1

n )‖T‖εα

.

Hence, for each n = 1, 2, . . . , there exists a unique un ∈ M such that

Sn(un) = un,

i.e.,

T (un +1nv) = λnun, (6.47)

whereλn = ‖T (un +

1nv)‖.

This implies that λn ≤ 2‖T‖. Since,

m(un, v)v ≤ un,

and m(un, v) > 0 is the maximal number λ such that λv ≤ un, we obtain that

un =1λnT (un +

1nv)

≥ 1λnT (m(un, v)v +

1nv)

≥ α

λn(m(un, v) +

1n

)v.

Therefore, by the maximality of m(un, v), we obtainα

λn

(m(un, v) +

1n

)≤ m(un, v),

i.e.,

α(1 +

1nm(un, v)

)≤ λn.

The sequence λn is therefore uniformly bounded away from zero and, as hasbeen shown above, also bounded above. It therefore has a convergent subsequence,λni, converging, say, to λ > 0. We now use equation (6.47) and the compactnessof T to obtain that the sequence uni has a convergent subsequence, converging,to, say, u, ‖u‖ = 1, and, since T is continuous,

T (u) = λu.


Since it must be the case that u ∈ K+, we see that, in fact, u ∈ intK, and hence,u ∈ M.

If u1, u2 ∈ M are such that

T (u1) = λ1u1, T (u2) = λ2u2,

thenu1 ≥ m(u1, u2)u2, m(u1, u2) > 0,

andλ1u1 = T (u1) ≥ m(u1, u2)T (u2) = m(u1, u2)λ2u2,

hence,

u1 ≥(m(u1, u2)

λ2

λ1

)u2.

On the other handm(u1, u2) = supα : αu2 ≤ u1,

which implies that λ1 ≥ λ2. Reversing the roles of u1 and u2, we obtain λ1 ≤ λ2,and thus,

λ1 = λ2 = λ.

Sinceu1 ≥ m(u1, u2)u2, m(u1, u2) > 0,

we haveT (u1 −m(u1, u2)u2) ∈ intK,

unlessu1 −m(u1, u2)u2 = 0.

On the other hand

T (u1 −m(u1, u2)u2) = λ(u1 −m(u1, u2)u2),

and thus, ifu1 −m(u1, u2)u2 ∈ K+,

thenu1 −m(u1, u2)u2 ∈ intK,

and we obtain a contradiction to the maximality of m(u1, u2). Hence,

u1 = m(u1, u2)u2

and since, ‖u1‖ = ‖u2‖ = 1, we have that m(u1, u2) = 1 and we have proved thatu1 = u2.

7. Integral equations

In this chapter, we shall present the basic existence and uniqueness theorem forsolutions of initial value problems for systems of ordinary differential equations. Weshall also discuss the existence of mild solutions of integral equations which underadditional assumptions provide the existence of solutions of initial value problemsfor parabolic partial differential equations. We conclude the chapter by presentingsome results about functional differential equations and integral equations.


7.1. Initial value problems. To this end let D be an open connected subset ofR× E, where E is a Banach space, and let

f : D → E

be a continuous and bounded mapping, i.e., it maps bounded sets in D to boundedsets in E.

We consider the differential equation

u′ = f(t, u), ′ =d

dt. (7.1)

and seek sufficient conditions for the existence of solutions of (7.1), where u ∈C1(I, E), with I an interval, I ⊂ R, is called a solution, if (t, u(t)) ∈ D, t ∈ I and

u′(t) = f(t, u(t)), t ∈ I.

By an initial value problem we mean the following:Given a point (t0, u0) ∈ D we seek a solution u of (7.1) defined on some open

interval I such thatu(t0) = u0, t0 ∈ I. (7.2)

We have the following proposition whose proof is straightforward:

Proposition 7.1. A function u ∈ C1(I, E), with I ⊂ R, and I an interval contain-ing t0 is a solution of the initial value problem (7.1), satisfying the initial condition(7.2) if, and only if, (t, u(t)) ∈ D, t ∈ I, and

u(t) = u(t0) +∫ t

t0

f(s, u(s))ds. (7.3)

The integral in (7.3) is a Riemann integral of a continuous function.We shall now, using Proposition 7.1, establish one of the classical and basic

existence and uniqueness theorems.

7.2. The Picard-Lindelof theorem. We say that f satisfies a local Lipschitzcondition on the domain D, provided for every closed and bounded set K ⊂ D,there exists a constant L = L(K), such that for all (t, u1), (t, u2) ∈ K

‖f(t, u1)− f(t, u2)‖ ≤ L‖u1 − u2‖,

where ‖ · ‖ is the norm in the space E. For such functions, one has the followingexistence and uniqueness theorem. This result is usually called the Picard-Lindeloftheorem.

Theorem 7.2. Assume that f : D → E is a continuous and bounded mappingwhich satisfies a local Lipschitz condition on the domain D. Then for every (t0, u0)in D, equation (7.1) has a unique solution on some interval I satisfying the initialcondition (7.2).

We remark that the theorem as stated is a local existence and uniqueness the-orem, in the sense that the interval I, where the solution exists will depend uponthe initial condition.

Proof. Let (t0, u0) ∈ D, then, since D is open, there exist positive constants a andb such that

Q = (t, u) : |t− t0| ≤ a, ‖u− u0‖ ≤ b ⊂ D.


Let L be the Lipschitz constant for f associated with the set Q. Further let

m := sup(t,u)∈Q

‖f(t, u)‖,

α := mina, bm.

Let L be any constant, L > L, and define I = [t0 − α, t0 + α], and

M := u ∈ C (I, E) : u(I) ⊂ Br(u0).

In C (I, E) we define a new norm as follows:

‖u‖M := max|t−t0|≤α

e−L|t−t0|‖u(t)‖.

And we let d(u, v) = ‖u− v‖M, then (M,d) is a complete metric space. Next definethe operator T on M by:

(Tu)(t) := u0 +∫ t

t0

f(s, u(s))ds, |t− t0| ≤ α. (7.4)

Then

‖(Tu)(t)− u0‖ ≤∣∣ ∫ t

t0

‖f(s, u(s))‖ds∣∣,

and, since u ∈ M,

‖(Tu)(t)− u0‖ ≤ αm ≤ b.

Hence T : M → M. Computing further, we obtain, for u, v ∈ M that

‖(Tu)(t)− (Tv)(t)‖ ≤∣∣∣ ∫ t

t0

‖f(s, u(s))− f(s, v(s))‖ds∣∣∣

≤ L∣∣∣ ∫ t

t0

‖u(s)− v(s)‖ds∣∣∣,

and hence

e−L|t−t0|‖(Tu)(t)− (Tv)(t)‖ ≤ e−L|t−t0|L∣∣∣ ∫ t

t0

‖u(s)− v(s)‖ds∣∣∣

≤ L

L‖u− v‖M.

Therefore,

d(Tu, Tv) ≤ L

Ld(u, v),

proving that T is a contraction mapping. The result therefore follows from thecontraction mapping principle.

We remark that, since T is a contraction mapping, the contraction mappingtheorem gives a constructive means for the solution of the initial value problem inTheorem 7.2 and the solution may in fact be obtained via an iteration procedure.This procedure is known as Picard iteration.


7.3. Abel-Liouville integral equations. In establishing the Picard-Lindelof the-orem we studied an associated integral equation

u(t) = u(t0) +∫ t

t0

f(s, u(s))ds. (7.5)

By translating the time variable appropriately and changing variables, there is noloss in generality in assuming that t0 = 0 in (7.5). In this section we shall consider ageneralization of this integral equation, namely an equation of Abel-Liouville type

u(t) = v(t) +1

Γ(µ)

∫ t

0

(t− s)µ−1f(t, s, u(s))ds, 0 ≤ t ≤ a, a > 0, (7.6)

where µ ∈ (0, 1], and Γ is the Gamma function. Choosing µ = 1 and v(t) =constant, one clearly obtains (7.5) as a special case.

While the result to follow will be valid for Banach space valued functions, we shallrestrict ourselves to the case of real-valued functions, remarking that the treatmentwill be similar in the more general setting. The discussion below follows the paperof Rainermann and Stallbohm [61], see also [42].

We shall introduce the following notations and make the assumptions:

(1) v : [0, a] → R is a continuous function.(2)

S := z : |v(s)− z| ≤ b, 0 ≤ s ≤ a,where b > 0 is a fixed positive constant.

(3)∆ := (t, s) : 0 ≤ s ≤ t ≤ a.

(4) f : ∆× S → R is a continuous function and

M := max|f(t, s, z)| : (t, s, z) ∈ ∆× S.

(5) for all (t, s, z1), (t, s, z2) ∈ ∆× S,

sµ|f(t, s, z1)− f(t, s, z2)| ≤ Γ(µ+ 1)|z1 − z2|. (7.7)

(6)

α := min(a,

(Γ(µ+ 1)

b

M

)1/µ).

(7)

M1 := w ∈ C[0, α] : w(0) = v(0), max0≤t≤α

|w(t)− v(t)| ≤ b.

(8) We define the operator T : M1 → C[0, α] by

T (w)(t) := v(t) +1

Γ(µ)

∫ t

0

(t− s)µ−1f(t, s, u(s))ds, 0 ≤ t ≤ α. (7.8)

Lemma 7.3. With the above notation and assumptions, we have

T : M1 → M1.

Proof. It is clear that T : M1 → C[0, α]. Further

T (w)(0) = v(0),


and for 0 ≤ t ≤ α,

|T (w)(t)− v(t)| ≤M1

Γ(µ)

∫ t

0

(t− s)µ−1ds

≤Mtµ

Γ(µ+ 1)

≤Mαµ

Γ(µ+ 1)≤ b.

We next letM := T (M1)

and find a metric d on M so that (M,d) becomes a complete metric space. Theabove lemma, of course, implies that T : M → M.

We next define d : M×M → [0,∞) by

d(w1, w2) := supt∈[0,α]

|w1(t)− w2(t)|.

Then, it is easily seen that d is a metric on M and, using the continuity assumptionsimposed on f , that for all u1, u2 ∈ M,

|T (u1)(t)− T (u2)(t)| ≤1

Γ(µ)

∫ t

0

(t− s)µ−1|f(t, s, u1(s))− f(t, s, u2(s))|ds

≤ tµ max0≤s≤t

|f(t, s, u1(s))− f(t, s, u2(s))|,

implying that (recall that u1(0) = u2(0))

t−µ|T (u1)(t)− T (u2)(t)| → 0.

Also

|T (u1)(t)− T (u2)(t)| ≤1

Γ(µ)

∫ t

0

(t− s)µ−1|f(t, s, u1(s))− f(t, s, u2(s))|ds

≤∫ t

0

(t− s)µ−1s−µ|u1(s)− u2(s)|ds.

On the other hand, if u1 6= u2, then d(u1, u2) > 0. Thus, if d(T (u1), T (u2)) 6= 0,we may choose t1 ∈ (0, α] so that

d(T (u1), T (u2)) = t−µ1 |T (u1)(t1)− T (u2)(t1)|

≤ Γ(µ+ 1)Γ(µ)

t−µ1

∫ t

0

(t− s)µ−1s−µ|u1(s)− u2(s))|ds

< µt−µ1

∫ t

0

(t− s)µ−1dsd(u1, u2),

where the latter strict inequality follows from the continuity of the functions in-volved and the fact that there exists s ∈ (0, t1) such that

s−µ|u1(s)− u2(s))| < d(u1, u2).

Using calculations and considerations like the above, it is straightforward toshow that the family M is an equicontinuous family of functions which is uniformlybounded. Hence, for a given u ∈ M the sequence of iterates Tn(u) will have a


convergent subsequence, say Tnj (u), converging to some function v ∈ M1. Thesequence Tnj+1(u), will therefore converge to T (v) ∈ M.

It therefore follows from Edelstein’s contraction mapping principle, TheoremIII.3.4, that equation (7.6) has a unique solution in the metric space M.

7.4. Mild solutions. Let E be a Banach space and let

S : [0,∞) → L(E,E)

(L(E,E) are the bounded linear maps from E to E) be a family of bounded linearoperators which form a strongly continuous semigroup of operators, i.e.,

S(t+ s) = S(t)S(s), ∀t, s ≥ 0,

S(0) = id, the identity mapping

limt→t0

S(t)x = S(t0)x, ∀t0 ≥ 0, x ∈ E.

For such semigroups it is the case that there exist constants (see [76]) β ∈ R andM > 0 such that

‖S(t)‖ ≤Meβt, t ≥ 0.(In this section, we shall use ‖·‖ for both the norm in E, and the norm in L(E,E).)

Let f : [0,∞) → E be a continuously differentiable function and let

u(t) := S(t)x+∫ t

0

S(t− s)f(s)ds. (7.9)

The integral in (7.9) is a Riemann integral of a continuous function. Then forx ∈ D(A), where

D(A) := x : Ax := limt→0+

1t

(S(t)x− x) exists,

(the operator A is called the infinitesimal generator of the semigroup S(t); t ≥ 0)u, given by (7.9) is the solution of the initial value problem (7.10) (see [69], [76])

du

dt= Au(t) + f(t), u(0) = x. (7.10)

On the other hand, if it is only assumed that f is continuous, then a solution of(7.9) need not necessarily be a solution of (7.10). One calls the function u definedby (7.9) a mild solution of (7.10). Thus a mild solution is defined for all x ∈ E,even if D(A) is a proper subset of E.

For example, one may easily verify that, if A ∈ L(E,E), then

S(t) := eAt =∞∑n=0

Antn

n!

is a strongly continuous semigroups of operators and if u is defined by (7.9), thenu solves (7.10) for any x ∈ E and any continuous f : [0,∞) → E.

We have the following theorem for the existence of mild solutions.

Theorem 7.4. Let S : [0,∞) → L(E,E) be a strongly continuous semigroup ofoperators and let

f : [0,∞)× E → E

be a continuous mapping which satisfies the Lipschitz condition

‖f(t, u)− f(t, v)‖ ≤ L‖u− v‖,∀t ∈ [0,∞), ∀u, v ∈ E, (7.11)


where L is a positive constant. Then the equation

u(t) = S(t)x+∫ t

0

S(t− s)f(s, u(s))ds (7.12)

has a unique continuous solution u : [0,∞) → E, i.e., the initial value problem

du

dt= Au(t) + f(t, u), u(0) = x, (7.13)

where A is the generator of S(t); t ≥ 0, has a unique mild solution.

Proof. Given any τ > 0, we shall prove the existence of a unique continuous solutionof (7.12) defined on the interval [0, τ ]. Since τ is chosen arbitrarily the result willfollow, observing that if u solves (7.12) on an interval [0, τ1] it will solve (7.12) onan interval [0, τ2], for any τ2 ≤ τ1.

Let us consider the space E := C([0, τ ], E) with norm

‖u‖τ := max[0,τ ]

e−(L+β)t‖u(t)‖,

where L > 0 is to be chosen, and define the mapping T : E → E by

T (u)(t) := S(t)x+∫ t

0

S(t− s)f(s, u(s))ds. (7.14)

For u, v ∈ E we compute

e−(L+β)t‖T (u)(t)− T (v)(t)‖ ≤MLe−Lt∫ t

0

eLse−(L+β)s‖u(s)− v(s)‖ds.

It follows that

‖T (u)− T (v)‖τ ≤ML

L‖u− v‖τ . (7.15)

Therefore T is a contraction mapping providedML

L< 1. (7.16)

We choose L this way and obtain the existence of a unique fixed point u of T inE .

7.5. Periodic solutions of linear systems.

Mild periodic solutions. Let f : [0,∞) → E be a continuous function which isperiodic of period τ > 0, i.e.

f(t+ τ) = f(t), t ≥ 0.

In this section we show that the integral equation

u(t) = S(t)x+∫ t

0

S(t− s)f(s)ds, (7.17)

where S(t) : t ≥ 0 is a strongly continuous semigroup of operators, with infin-itesimal generator A, has a unique periodic solution, provided the semigroup is aso-called asymptotically stable semigroup. I.e., we shall show that the problem

du

dt= Au(t) + f(t), (7.18)

has a unique mild solution which is also periodic of period τ .


To this end, we shall employ the contraction mapping principle as given byTheorem 3.6 of Chapter 2.

We call the semigroup S(t) : t ≥ 0 asymptotically stable, provided there existconstants M > 0, β > 0 such that

‖S(t)‖ ≤Me−βt, t ≥ 0.

Well known examples of such asymptotically stable semigroups are given in thefinite dimensional case by eAt, where A is an N×N matrix all of whose eigenvalueshave negative real parts (see e.g. [36]), or in the infinite dimensional case by certainparabolic partial differential equations (see e.g. [29], [69]).

Let us define the operator T : E → E by

T (x) := S(τ)x+∫ τ

0

S(τ − s)f(s)ds. (7.19)

It then follows from Theorem 7.4 and the periodicity of the function f , that

Tn(x) = S(nτ)x+∫ nτ

0

S(nτ − s)f(s)ds, (7.20)

for any positive integer n. We have

‖Tn(x)− Tn(y)‖ = ‖S(nτ)x− S(nτ)(y)‖≤ ‖S(nτ)‖‖x− y‖

≤Me−βnτ‖x− y‖, ∀x, y ∈ E.(7.21)

SinceMe−βnτ < 1,

for n, large enough, we have that Tn is a contraction mapping for such n. It followstherefore that Tn, hence T , has a unique fixed point x ∈ E. The solution

u(t) = S(t)x+∫ t

0

S(t− s)f(s)ds, (7.22)

is a periodic function of period τ .Summarizing the above, we have proved the following theorem.

Theorem 7.5. Let S(t) : t ≥ 0 be an asymptotically stable, strongly continuoussemigroup with infinitesimal generator A. Then for any continuous f : [0,∞) → Ewhich is periodic of period τ > 0, there exists a unique mild periodic solution u, ofperiod τ of equation (7.18).

7.6. The finite dimensional case. We consider next the system of equations

du

dt= Au(t) + f(t), (7.23)

where A is an N ×N matrix and f : R → RN , is a function of period T .We shall assume here that A is a matrix all of whose eigenvalues have nonzero

real part. If this is the case then there exists a nonsingular matrix P such that(see, e.g. [70])

P−1AP =(A1 OO A2

),

where A1 is an N1 ×N1 and A2 is an N2 ×N2 matrix with N1 +N2 = N , and alleigenvalues of A1 have negative real parts and those of A2 have positive real parts,


and the matrices O are zero matrices of appropriate dimensions. Hence if we makethe transformation u = Pv, the the system (7.23 becomes

dv

dt= P−1APv(t) + P−1f(t). (7.24)

This is a decoupled system which we may rewrite as

dv1dt

= A1v1(t) + f1(t)

dv2dt

= A2v2(t) + f2(t),(7.25)

where

v =(v1v2

), P−1f =

(f1f2

).

On the other hand v is a solution of (7.25) if and only if

w =(w1(t)w2(t)

)=

(v1(t)v2(−t)

)is a solution of

dw1

dt= A1v1(t) + f1(t)

dw2

dt= −A2w2(t)− f2(−t).

(7.26)

The fundamental solution S(t) of the system (7.26) is given by

S(t) = eBt,

where B is the matrix

B =(A1 OO −A2

),

all of whose eigenvalues have negative real part. Hence there exist constants M > 0,β > 0, such that

‖S(t)‖ ≤Me−βt.

We may therefore apply Theorem 7.5 to conclude that equation (7.26) and hence(7.23) have unique periodic solutions.

We remark here that the existence of a unique periodic solution of (7.23) alsoeasily follows from the fact that A is assumed not to have any eigenvalues with zeroreal part, and hence that

id−eAT ,

where id is the N ×N identity matrix, is nonsingular for any T 6= 0. Furthermorethe reduction made above shows that, without loss in generality we may assumethat all the eigenvalues of A have negative real parts, say

Re(λi) ≤ −β < 0, (7.27)

where β is a a positive constant and λ1, . . . , λN are the eigenvalues of A. Usinga further change of basis (using the Jordan canonical form of A) we may assume


that A has the form (the, perhaps, unconventional labeling has been chosen forconvenience’s sake)

A =

λN a2,N , . . . , aN,N0 λN−1, . . . aN−1,N

.... . .

......

0 . . . . . . λ1

.

We thus consider the system

du

dt= Au(t) + f(t), (7.28)

where

u =

uNuN−1

...u1

, f =

fNfN−1

...f1

.

If then u is a T− periodic solution of (7.28) it must be the case that

u1(t) = eλ1t(c+

∫ t

0

e−λ1sf1(s)ds).

Since u1 is periodic, it must be bounded on the real line and hence,

c+∫ t

0

e−λ1sf1(s)ds→ 0, as t→ −∞;

i.e.,

c = −∫ −∞

0

e−λ1sf1(s)ds

or

u1(t) =∫ t

−∞eλ1(t−s)f1(s)ds.

Therefore,

|u1(t)| ≤1β‖f1‖,

(see (7.27) for the choice of β) where

‖f1‖ = supt∈R

|f1(t)|.

We next consider the component u2 of u. Since u1 has been found (and estimated),we can employ a similar argument and the estimate on u1 to find that

|u2(t)| ≤1β‖a2,Nu1 + f2‖,

and thus|u2(t)| ≤

c2β

max‖f1‖, ‖f2‖,

where the constant c2 only depends upon the matrix A. Using an induction argu-ment one obtains for i = 1, . . . , N

|ui(t)| ≤ciβ

max‖f1‖, . . . , ‖fi‖


with constants ci depending on A, only. We therefore have, letting

‖u‖ = maxi=1,...,N

‖ui‖, ‖f‖ = maxi=1,...,N

‖fi‖,

that‖u‖ ≤ c

β‖f‖,

where c is a constant, depending upon A only. We summarize the above in thefollowing theorem.

Theorem 7.6. Let A be a matrix none of whose eigenvalues has zero real part.Then for any given forcing term f , a periodic function of period T , there existsa unique periodic solution u of (7.28). Further, there exists a constant c whichdepends upon A only, such that

‖u‖ ≤ c

β‖f‖,

with the norms defined above.

We note that the conclusion of Theorem 7.6 is equally valid if we replace therequirement that f be periodic with the requirement that f be bounded.

7.7. Almost periodic differential equations. In this section we shall return tothe equation

du

dt= Au(t) + f(t), (7.29)

and the more general nonlinear equation

du

dt= Au(t) + f(t, u), (7.30)

where it is assumed that A is an N ×N matrix and either

f : R → RN or f : R× RN → RN

is a function which is almost periodic in the t variable (see definitions below). It willbe assumed that all eigenvalues of A have nonzero real part which will imply thatfor almost periodic f equation (7.29) has a unique almost periodic solution. Thisfact will be employed for the study of equation (7.30) under the assumption thatf satisfies a Lipschitz condition with respect to the dependent variable, in whichcase (7.30) will be shown to have a unique almost periodic solution, provided theLipschitz constant of f is small enough. Our presentation relies mainly on the workof Coppel [21], see also [22], [4] and [31], and Theorem 7.6, above.

7.8. Bounded solutions. We consider again the system (7.28) under the assump-tion that none of the eigenvalues of A have zero real part and f a continuous(not necessarily periodic) function. It then follows from the superposition principlethat, if (7.29) has a bounded solution, no other bounded solutions may exist (theunperturbed system

u′ = Au,

has the zero solution as the only bounded solution). Furthermore the discussionabove, proving Theorem 7.6, may be used to establish the following result.


Theorem 7.7. Let A be a matrix none of whose eigenvalues has zero real part.Then for any given forcing term f , with f bounded on R, there exists a uniquebounded solution u of (7.29). Further, there exists a constant c which depends uponA only, such that

‖u‖ ≤ c

β‖f‖,

with the norms as defined before.

7.9. Almost periodic functions. Let us denote by V the set of continuous func-tions

V := f : R → RN : ∃T > 0 : f(t+ T ) = f(t), t ∈ R;i.e., the set of all continuous functions which are periodic of some period, the periodnot being fixed. We then define

V := span(V ),

i.e., the smallest vector space containing V . We note that for f ∈ V ,

‖f‖ := supt∈R

maxi=1,...,N

|fi(t)| <∞,

and that ‖ · ‖ defines a norm in V . We then denote by E the completion of V withrespect to this norm (the norm of uniform convergence on the real line). This spaceis the space of almost periodic functions which, by definition, includes all periodicand quasiperiodic functions (i.e., finite linear combinations of periodic functionshaving possibly different periods). It follows that an almost periodic function is theuniform limit (uniform on the real line) of a sequence of quasiperiodic functions.

For detailed studies of almost periodic functions we refer the reader to [4], [12],[30], [22], and [31].

7.10. Almost periodic systems. In this section we shall first establish an ex-tension to almost periodic systems of Theorem 7.5 and then use it to establish theexistence of a unique almost periodic solution of system (7.30), in case the nonlinearforcing term f satisfies a Lipschitz condition with respect to the dependent variableu. We have the following theorem.

Theorem 7.8. Let A be a matrix none of whose eigenvalues has zero real part.Then for any given almost periodic forcing term f , there exists a unique almostperiodic solution u of (7.29). Further, there exists a constant c which depends uponA only, such that

‖u‖ ≤ c

β‖f‖,

with the norms defined above.

Proof. Let us consider system (7.29) in case the forcing term f is quasiperiodic. Inthis case f may be written as a finite linear combination of periodic functions, say

f(t) =k∑i=1

fi(t),

where fi has period Ti, i = 1, . . . , k. It follows from Theorem 7.5 that each of thesystems

u′ = Au+ fi(t),


has a unique periodic solution ui(t), of period Ti, i = 1, . . . , k. Hence, the super-position principle and Theorem 7.6 imply that (7.30) has the unique quasiperiodicsolution

u(t) =k∑i=1

ui(t),

and‖u‖ ≤ c

β‖f‖.

On the other hand, if f is an almost periodic function, there exists a sequence ofquasiperiodic functions fn(t) such that

f(t) = limn→∞

fn(t),

where the limit is uniform on the real line. We let un(t) be the sequence ofquasiperiodic solutions of (7.29) with f replaced by fn, n = 1, 2, . . . . Then

‖un − um‖ ≤c

β‖fn − fm‖.

Hence, un(t) is a Cauchy sequence of quasiperiodic functions which is uniform onthe real line and therefore must converge to an almost periodic function u. Usingequivalent integral equations, as in Section 1, one shows that u solves (7.29) and,since u is bounded it must satisfy

‖u‖ ≤ c

β‖f‖.

We next consider the problem (7.30). We have the following theorem:

Theorem 7.9. Let A be a matrix none of whose eigenvalues has zero real part. Let

f : R× RN → RN

be such that

|f(t, u)− f(t, v)| ≤ L|u− v|, ∀t ∈ R, ∀u, v ∈ RN (7.31)

and let f(·, u) be almost periodic for each u ∈ RN . Then there exists a constant L0

such that for eachL < L0,

equation (7.30 has a unique almost periodic solution u.

Proof. It follows from the Weierstrass approximation theorem and the definitionof almost periodicity (see e.g. [22], [21]) that for each almost periodic function vthe composition f(t, v(t)) is an almost periodic function. We may thus define anoperator T : E → E, v 7→ T (v) by setting

u := T (v),

where u is the unique almost periodic solution of

u′ = Au+ f(t, v(t)),

whose existence follows from Theorem 7.8. This theorem also implies that

|T (v)(t)− T (w)(t)| ≤ c

βsupt∈R

|f(t, v(t))− f(t, w(t))|,


i.e., using the Lipschitz condition satisfied by f ,

‖T (v)− T (w)‖ ≤ Lc

β‖v − w‖.

The operator T will therefore be a contraction mapping, provided that

Lc

β< 1.

Thus, if we choose

L0 =β

cthe contraction mapping principle will apply and the existence of a unique almostperiodic solution follows.

8. The implicit function theorem

In this chapter we shall prove the implicit function theorem for mappings definedbetween Banach spaces which are Frechet differentiable.

8.1. Frechet differentiable mappings. Let us assume we have Banach spacesE,X and let

f : U → X,

(where U is open in E) be a continuous mapping. Let u0 ∈ U , then f is said to beFrechet differentiable at u0 provided there exists

L ∈ L(E,X)

(the continuous (or bounded) linear mappings from E to X) such that

f(u0 + v) = f(u0) + L(v) + o(‖v‖).If f is Frechet differentiable for every u ∈ U , the mapping U → L(E,X) given

byu 7→ Duf(u),

where Duf(u) is the Frechet derivative of f at u, is then defined. We remark, thatthe Frechet derivative is uniquely determined (if it exists) and the above definitionprovides a Taylor expansion expression. (See [37], [63].)

8.2. The implicit function theorem. Let us assume we have Banach spacesE,X,Λ and let

f : U × V → X,

(where U is open in E, V is open in Λ) be a continuous mapping satisfying thefollowing condition:

For each λ ∈ V the map f(·, λ) : U → X is Frechet-differentiableon U with Frechet derivative Duf(u, λ) and the mapping (u, λ) 7→Duf(u, λ) is a continuous mapping from U × V to L(E,X) (thecontinuous (or bounded) linear mappings from E to X).

Theorem 8.1. Let f satisfy the above condition and let there exist (u0, λ0) ∈ U×Vsuch that Duf(u0, λ0) is a linear homeomorphism of E onto X (i.e. Duf(u0, λ0) ∈L(E,X) and [Duf(u0, λ0)]−1 ∈ L(X,E)). Then there exist δ > 0, r > 0, andunique continuous mapping u : Bδ(λ0) = λ : ‖λ− λ0‖Λ ≤ δ → E such that

f(u(λ), λ) = f(u0, λ0), u(λ0) = u0 (8.1)


and‖u(λ)− u0‖ ≤ r, ∀λ ∈ Bδ(λ0).

Proof. Let us consider the equation

f(u, λ) = f(u0, λ0)

which is equivalent toT (f(u, λ)− f(u0, λ0)) = 0, (8.2)

where T = [Duf(u0, λ0)]−1, or

u = u− T (f(u, λ)− f(u0, λ0))=:G(u, λ). (8.3)

The mapping G has the following properties:(i) G(u0, λ0) = u0,(ii) G and DuG are continuous in (u, λ),(iii) DuG(u0, λ0) = 0.

Hence, since

G(u1, λ)−G(u2, λ) =∫ 1

0

DuG (u2 + t(u1 − u2), λ) (u1 − u2)dt,

we obtain

‖G(u1, λ)−G(u2, λ)‖ ≤(

sup0≤t≤1

‖DuG(u1 + t(u2 − u1), λ)‖L)‖u1 − u2‖

≤ 12‖u1 − u2‖,

(8.4)

provided ‖u1 − u0‖ ≤ r, ‖u2 − u0‖ ≤ r, ‖λ− λ0‖Λ ≤ δ, where r > 0 and δ > 0 aresmall enough. Now

‖G(u, λ)− u0‖ = ‖G(u, λ)−G(u0, λ0)‖≤ ‖G(u, λ)−G(u0, λ)‖+ ‖G(u0, λ)−G(u0, λ0)‖

≤ 12‖u− u0‖+ ‖G(u0, λ)−G(u0, λ0)‖

≤ 12r +

12r,

provided ‖λ− λ0‖Λ ≤ δ, where δ > 0 has been further restricted so that

‖G(u0, λ)−G(u0, λ0)‖ ≤12r.

We now think of u as a continuous function

u : Bδ(λ0) → E

and define

M :=u : Bδ(λ0) → E, such that u is continuous,

u(λ0) = u0, u(Bδ(λ0)) ⊂ Br(u0),

and equip M with the norm

‖u‖M := supλ∈Bδ(λ0)

‖u(λ)‖.


Then M is a closed subset of the Banach space of bounded continuous functionsdefined on Bδ(λ0) with values in E. Since E is a Banach space, it is a completemetric space. Thus, (8.3) defines an equation

u(·) = G(u(·), ·) (8.5)

in M.Define g by (here we think of u as an element of M)

g(u)(λ) := G(u(λ), λ),

then g : M → M and it follows from (8.4) that

‖g(u)− g(v)‖M ≤ 12‖u− v‖M,

hence g has a unique fixed point by the contraction mapping principle (Theorem3.1 of Chapter 2).

Remark 8.2. If in the implicit function theorem f is k times continuously differ-entiable with respect to λ, then the mapping λ 7→ u(λ) inherits this property.

Proof. We sketch a proof for the case that f is continuously differentiable withrespect to λ. It follows from the above computation that

‖DuG(u(λ), λ)‖L ≤12,

for ‖λ− λ0‖Λ ≤ δ. It follows that

[id−DuG(u(λ), λ)]−1DλG(u(λ), λ) ∈ L(Λ, E).

Furthermore, a quick calculation shows that

u(λ+ h) = u(λ) + [id−DuG(u(λ), λ)]−1DλG(u(λ), λ)(h) + o(‖h‖Λ)

and thusDλu(λ) = [id−DuG(u(λ), λ)]−1

DλG(u(λ), λ)is continuous with respect to the parameter λ.

8.3. Two examples.

Example 8.3. Let us consider the nonlinear boundary value problem

u′′ + λeu = 0, 0 < x < π, u(0) = 0 = u(π). (8.6)

This is a one space-dimensional mathematical model from the theory of combustion(cf. [8]), where u represents a dimensionless temperature.

The problem may be explicitly solved using quadrature methods and was firstposed by Liouville [52]. We shall show, by an application of Theorem 8.1, thatfor λ ∈ R, in a neighborhood of 0, (8.6) has a unique solution of small norm inC2([0, π]). To this end we define

E := C20 ([0, π]) := C2([0, π]) ∩ u : u(0) = 0 = u(π)

X := C([0, π]), Λ := R.These spaces are Banach spaces when equipped with their usual norms, i.e.,

‖u‖X := supt∈[0,π]

|u(t)|,

‖u‖E := ‖u‖X + ‖u′‖X + ‖u′′‖X .


and | · | represents absolute value.We let f : E × Λ → X be given by

f(u, λ) := u′′ + λeu.

Then f is continuous and f(0, 0) = 0. (When λ = 0 (no heat generation) the uniquesolution is u ≡ 0.) Furthermore, for u0 ∈ E, Duf(u0, λ) is given by (the readershould carry out the verification)

Duf(u0, λ)v = v′′ + λeu0v,

and, hence, the mapping(u, λ) 7→ Duf(u, λ)

is continuous. Let us consider the linear mapping

T := Duf(0, 0) : E → X.

We must show that this mapping is a linear homeomorphism. To see this we notethat for every h ∈ X, the unique solution of

v′′ = h(x), 0 < x < π, v(0) = 0 = v(π),

is given by

v(x) =∫ π

0

G(x, s)h(s)ds, (8.7)

where

G(x, s) =

− 1π (π − x)s, 0 ≤ s ≤ x

− 1πx(π − s), x ≤ s ≤ π.

(8.8)

From the representation (8.7) we may conclude that there exists a constant c suchthat

‖v‖E = ‖T−1h‖E ≤ c‖h‖X ;i.e. T−1 is one to one and continuous. Hence, all conditions of the implicit functiontheorem are satisfied and we may conclude that for each sufficiently small λ, (8.6)has a unique small solution u ∈ C2([0, π].), Furthermore, the map λ 7→ u(λ) iscontinuous (in fact, smooth) from a neighborhood of 0 ∈ R to C2([0, π]). Weobserve that this ‘solution branch’ (λ, u(λ)) is bounded in the λ− direction. To seethis, we note that if λ > 0 is such that (8.6) has a solution, then the correspondingsolution u must be positive, u(x) > 0, 0 < x < π. Hence

0 = u′′ + λeu > u′′ + λu. (8.9)

Let v(x) = sinx. Then v satisfies

v′′ + v = 0, 0 < x < π, v(0) = 0 = v(π). (8.10)

From (8.9) and (8.10) we obtain

0 >∫ π

0

(u′′v − v′′u)dx+ (λ− 1)∫ π

0

uvdx,

and, hence, integrating by parts,

0 > (λ− 1)∫ π

0

uvdx,

implying that λ < 1.As a second example, we consider the following.


Example 8.4. Given any l0 6= n2, n = 1, 2, . . . , the forced nonlinear oscillator(periodic boundary value problem)

u′′ + lu+ u2 = g, u(0) = u(2π), u′(0) = u′(2π) (8.11)

where g is a continuous 2π − periodic function and l ∈ R, is a parameter, has aunique 2π periodic solution for all g of sufficiently small norm and |l−l0| sufficientlysmall.

Let

E := C2([0, 2π]) ∩ u : u(0) = u(2π), u′(0) = u′(2π),X := C([0, 2π]),

where both spaces are equipped with the norms as in the previous example. As aparameter space we choose

Λ := R×X.

The norm in Λ is given by ‖λ = (l, g)‖Λ = |l| + ‖g‖X . We shall show that, forcertain values of l, (8.11) has a unique solution for all forcing terms g of smallnorm.

To this end let f : E × Λ → X be given by

f(u, λ) = f(u, l, g) := u′′ + lu+ u2 − g.

Then f(0, 0) = 0, and Duf(u, λ) is defined by

(Duf(u, λ))(v) = v′′ + lv + 2uv,

and hence the mappingu 7→ Duf(u, λ)

is a continuous mapping of E to L(E;X), i.e. f is a C1 mapping. It follows fromelementary differential equations theory (see e.g., [11]) that the problem

v′′ + l0v = h,

has a unique 2π–periodic solution for every 2π–periodic h as long as l0 6= n2,n = 1, 2, . . . , and that ‖v‖ ≤ C‖h‖X for some constant C (only depending uponl0). Hence, Duf(0, l0, 0) is a linear homeomorphism of E onto X whenever l0 6= n2,n = 1, 2, . . . , and we conclude that for every g ∈ X of small norm and |l − l0|sufficiently small, (8.11) has a unique solution u ∈ E of small norm.

We note that the above example is prototypical for forced nonlinear oscillators.Virtually the same arguments can be applied (the reader might carry out the nec-essary calculations) to conclude that the forced pendulum equation

u′′ + l sinu = g

has a unique 2π- periodic response of small norm for every 2π - periodic forcingterm g of small norm, as long as l 6= n2, n = 1, 2, . . . .

9. Variational inequalities

In this chapter we shall discuss existence results for solutions of variational in-equalities which are defined by bilinear forms on a Banach space. The main resultproved is a Lax-Milgram type result. The approach follows the basic paper ofLions-Stampacchia [51] and also [44].


9.1. On symmetric bilinear forms. Let E be a real reflexive Banach space withnorm ‖ · ‖ and let

a : E × E → Rbe a continuous, coercive, symmetric, bilinear form, i.e.,

|a(u, v)| ≤ c1‖u‖‖v‖, a(u, u) ≥ c2‖u‖2, a(u, v) = a(v, u), ∀u, v ∈ E,and a is linear in each variable separately, where c1 and c2 are positive constants.As is common, we denote by E∗ the dual space of E and for b ∈ E∗ we denoteby 〈b, u〉 the value of the continuous linear functional b at the point u, the pairingbetween E∗ and E. The norm in E∗ we shall denote by ‖ · ‖∗.

Along with the norm topology on E, we shall also have occasion to make use ofthe weak topology (see below and, e.g., [62], [71], [76]).

For given b ∈ E∗ and a weakly closed set K we consider the functional

f(u) =12a(u, u)− 〈b, u〉. (9.1)

An easy computation shows that

f(u) ≥ c22‖u‖2 − ‖b‖E∗‖u‖

and, hence, thatf(u) →∞, as ‖u‖ → ∞

(f is coercive) and that f is bounded below on E. Hence, it is the case that

α := infv∈K

f(v) > −∞.

Let us choose a sequence un∞n=1 in K (a minimizing sequence) such that

f(un) → α.

It follows (because f is coercive) that the sequence un∞n=1 is a bounded sequenceand hence has (since E is reflexive, see [62], [71], [76]) a weakly convergent subse-quence, converging weakly to, say, u. We denote this subsequence, after appropriaterelabeling, again by un∞n=1 and hence have

un u

( denotes weak convergence), i.e. for any element h ∈ E∗

〈h, un〉 → 〈h, u〉.Since K is weakly closed, we have that u ∈ K. Since a is bilinear and nonnegative(a is coercive), we obtain

a(un, un) ≥ a(un, u) + a(u, un)− a(u, u),

and thatlim infn→∞

a(un, un) ≥ a(u, u)

(the form is weakly sequentially lower semicontinuous). We may, therefore, concludethat

f(u) = minv∈K

f(v). (9.2)

Let us now assume that the set K is also convex (hence, it is also closed, since,in reflexive Banach spaces, convex sets are closed if, and only if, they are weaklyclosed, cf. [71], [76]). Then for any v ∈ K and 0 ≤ t ≤ 1 we have that

f(u) ≤ f(u+ t(v − u)). (9.3)


Computing f(u+ t(v−u))−f(u), and using the properties of the form a, we obtain

0 ≤ ta(u, v − u)− t〈b, v − u〉+ t2a(v − u, v − u), 0 ≤ t ≤ 1, v ∈ K.Hence, upon dividing the latter inequality by t > 0, and letting t→ 0, we see thatthere exists u ∈ K such that

f(u) = minv∈K

f(v), (9.4)

a(u, v − u) ≥ 〈b, v − u〉, ∀v ∈ K. (9.5)

Hence, if b1, b2 ∈ E∗ are given and u1, u2 are solutions in K of the correspondingproblems (9.4), (9.5), then, denoting by

Tbi = ui, i = 1, 2,

we easily conclude from (9.5) and the coerciveness of a that

‖Tb1 − Tb2‖ ≤1c2‖b1 − b2‖∗. (9.6)

Thus, we see that the problems (9.4), (9.5) have a unique solution.We have the following theorem.

Theorem 9.1. Let a : E × E → R be a continuous, bilinear, symmetric, andcoercive form and let K be a closed convex subset of E. Then for any b ∈ E∗ thevariational inequality

a(u, v − u) ≥ 〈b, v − u〉, ∀v ∈ K. (9.7)

has a unique solution u ∈ K. Hence, equation (9.7) defines a solution mapping

T : E∗ → K, b 7→ Tb = u,

which is Lipschitz continuous with Lipschitz constant 1c2

, where c2 is the coercivityconstant of a.

Remark 9.2. It follows from the above considerations that if a satisfies the condi-tions of Theorem 9.1 except that it is not necessarily symmetric, and if inequality(9.7) has a solution for every b ∈ E∗, then the solution mapping T , above is well-defined and satisfies the Lipschitz condition (9.6).

9.2. Bilinear forms continued. Let a : E × E → R be a continuous, coercive,bilinear form, b ∈ E∗, and K a closed convex set.

9.3. The problem. We pose the following problem: Find (prove the existence of)u ∈ K such that

a(u, v − u) ≥ 〈b, v − u〉, ∀v ∈ K. (9.8)In case a is symmetric, this problem has been solved above and Theorem 9.1 pro-vides its solution. Thus, it remains to be shown that the theorem remains true incase a is not necessarily symmetric.

The development in this section follows closely the development in [51] and [44].

Uniqueness of the solution. Using properties of bilinear forms, one concludes(see Remark 9.2) that for all b ∈ E∗, problem (9.8) has at most one solution and ifb1, b2 ∈ E∗ and solutions u1, u2 exist, then

‖u1 − u2‖ ≤1c2‖b1 − b2‖∗,

where c2 is a coercivity constant of a.


Existence of the solution. We write a = ae + ao, where

ae(u, v) :=12(a(u, v) + a(v, u)),

ao(u, v) :=12(a(u, v)− a(v, u)),

then ae is a continuous, symmetric, coercive, bilinear form and ao is continuousand bilinear.

Consider the family of problems

ae(u, v − u) + tao(u, v − u) ≥ 〈b, v − u〉, ∀v ∈ K, 0 ≤ t ≤ 1, (9.9)

and let us denote byat := ae + ta0.

We have the following lemma.

Lemma 9.3. Let t ∈ [0,∞) be such that the problem

at(u, v − u) ≥ 〈b, v − u〉, ∀v ∈ K, (9.10)

has a unique solution for all b ∈ E∗. Then there exists a constant c > 0, dependingonly on the continuity and coercivity constants of a, such that problem

at+τ (u, v − u) ≥ 〈b, v − u〉, ∀v ∈ K, (9.11)

has a unique solution for all b ∈ E∗ and 0 ≤ τ ≤ c.

Proof. For w ∈ K and t ≥ 0, consider

at(u, v − u) ≥ 〈b, v − u〉 − τao(w, v − u), ∀v ∈ K. (9.12)

Note that for fixed w ∈ K,

bw := b− τao(w, ·) ∈ E∗,hence, there exists a unique u = Tw solving (9.12) and

‖Tw1 − Tw2‖ ≤1c2‖bw1 − bw2‖∗.

On the other hand

‖bw1 − bw2‖∗ = sup‖u‖=1

τ |ao(w1, u)− ao(w2, u)| ≤ τc1‖w1 − w2‖,

and hence‖Tw1 − Tw2‖ ≤

τc1c2‖w1 − w2‖,

and T : K → K is a contraction mapping provided τc1c2

< 1. Therefore, there isa unique solution of (9.11) as long as τ < c2

c1, and we may choose c = c2

2c1, for

example.

We may apply the above lemma with t = 0, since a0 = ae, and ae is symmetric,and obtain that

at(u, v − u) ≥ 〈d, v − u〉, ∀v ∈ K (9.13)has a unique solution for all d ∈ E∗, for 0 ≤ t ≤ c. Hence by the lemma, we obtainthat (9.13) has a unique solution for 0 ≤ t ≤ 2c, and continuing in this manner weobtain a unique solution of (9.13) for all t ∈ [0,∞), and in particular for t = 1, andwe have shown that problem (9.8) is uniquely solvable.

We therefore have the following theorem.


Theorem 9.4. Let a : E × E → R be a continuous, bilinear, and coercive formand let K be a closed convex subset of E. Then for any b ∈ E∗ the variationalinequality

a(u, v − u) ≥ 〈b, v − u〉, ∀v ∈ K. (9.14)has a unique solution u ∈ K. Hence equation (9.7) defines a solution mapping

T : E∗ → K, b 7→ Tb = u,

which is Lipschitz continuous with Lipschitz constant 1c2

, where c2 is a coercivityconstant of a.

Using this result one may immediately obtain an existence result for solutionsof nonlinearly perturbed variational inequalities of the form

a(u, v − u) ≥ 〈F (u), v − u〉, ∀v ∈ K, (9.15)

where F : E → E∗, is a Lipschitz continuous mapping, say,

‖F (u1)− F (u2)‖∗ ≤ k‖u1 − u2‖.We have the following result.

Theorem 9.5. Let a,K, F be as above. Then the variational inequality (9.15) hasa unique solution, provided that

k < c2,

where k is the Lipschitz constant for F and c2 is the coercivity constant for a.

Proof. It follows from Theorem 9.4 that the variational inequality (9.15) is equiv-alent to the fixed point problem

u = TF (u). (9.16)

SinceTF : K → K,

and K is a closed convex subset of E, hence a complete metric space, the result thenfollows from the contraction mapping principle and Theorem 9.4, once we observethat for any u1, u2 ∈ K

‖TF (u1)− TF (u2)‖ ≤1c2‖F (u1)− F (u2)‖∗ ≤

k

c2‖u1 − u2‖.

9.4. Some examples.

An obstacle problem. Let Ω be a bounded domain in RN and let E = L2(Ω).Let ψ ∈ E be given and let

K := u ∈ E : u(x) ≥ ψ(x), a.e. in Ω.Then K is a closed, convex subset of E. We let a : E × E → R be defined by

a(u, v) :=∫

Ω

uvdx = 〈u, v〉.

Thena(u, u) = ‖u‖2,

where ‖·‖ is the norm in the space E. Thus we see that a is a continuous, symmetric,coercive, and bilinear form.


Let b ∈ E and define f : E → R as

f(u) =12‖u‖2 − 〈b, u〉.

Then there exists a unique u ∈ K such that

f(u) = minv∈K

f(v)

and, furthermore, u solves the variational inequality

a(u, v − u)− 〈b, v − u〉 ≥ 0, ∀v ∈ K;

i.e., ∫Ω

(u− b)(v − u)dx ≥ 0, ∀v ∈ K, (9.17)

and the latter must have a unique solution. The natural candidate for this solutionis u = max(ψ, b), as one easily verifies by substituting into (9.17).

Another example. Let E := L2(0, 1), K := u :∫ 1

0udx = 1. Then K is closed

and convex (hence weakly closed). Let

a(u, v) :=∫ 1

0

uv dx = 〈u, v〉,

then, as above, a is a continuous, coercive, symmetric, and bilinear form. Hencethere exists a unique u ∈ K such that

a(u, v − u) ≥ 0, ∀v ∈ K,

i.e.

〈u, v − u〉 ≥ 0, ∀v ∈ K,

or

〈u, v〉 ≥ 〈u, u〉, ∀v ∈ K,

i.e. ∫ 1

0

uvdx ≥∫ 1

0

u2dx, ∀v ∈ K.

On the other hand ∣∣∣ ∫ 1

0

udx∣∣∣ ≤ ( ∫ 1

0

u2dx)1/2

,

and hence ∫ 1

0

uvdx ≥ 1, ∀v ∈ K.

Clearly u = 1 solves the inequality.


9.5. A second order boundary value problem. In this section we shall providean example of a boundary value problem for a second order ordinary differentialequation on the interval (0,∞) which may be solved by the methods developed here.It furnishes an example where the associated quadratic form is not symmetric.

Let us denote by E = H10 (0,∞) (the closure of the space C∞0 (0,∞) in the Sobolev

space of functions u : (0,∞) → R which together with their first distributionalderivatives are square integrable on (0,∞); see Chapter 2, section 2.3). The normin E is given by

‖u‖2 :=∫ ∞

0

u2dx+∫ ∞

0

(u′)2dx.

Let the quadratic form a : E × E → R, be given by

a(u, v) :=∫ ∞

0

u′v′dx+∫ ∞

0

uv′dx+∫ ∞

0

uvdx. (9.18)

One quickly may check that a is continuous, bilinear, and coercive (with coercivityconstant 1

2 ) but, it is clearly not symmetric. It follows that for any b ∈ L2(0,∞)the variational inequality

a(u, v − u)− 〈b, v − u〉 ≥ 0, ∀v ∈ E, (9.19)

has a unique solution, and, hence, since in this problem the convex set K is thewhole space E, the equation

a(u, v)− 〈b, v〉 = 0, ∀v ∈ E, (9.20)

has a unique solution. I.e., there exists a unique u ∈ E such that∫ ∞

0

u′v′dx+∫ ∞

0

uv′dx+∫ ∞

0

uvdx =∫ ∞

0

bvdx, ∀v ∈ E. (9.21)

Since C∞0 (0,∞) (the infinitely smooth functions with compact support, or testfunctions) is dense in E, we may interpret (9.21) in the sense of distributions andobtain

−∂2u(v)− ∂u(v) +∫ ∞

0

uvdx =∫ ∞

0

bvdx, ∀v ∈ C∞0 (0,∞), (9.22)

and hence−∂2u− ∂u+ u = b, (9.23)

has a unique solution u ∈ E for any b ∈ L2(0,∞) (here ∂2u, ∂u are the second,respectively, first distributional derivatives of the function u (see again Chapter 2,section 2.3)).

9.6. An obstacle problem. Let us consider here once more the quadratic form aof the previous section and pose the obstacle problem

a(u, v − u)− 〈b, v − u〉 ≥ 0, ∀v ∈ K, (9.24)

where K is the closed convex set

K := u ∈ H10 (0,∞) : u(x) ≥ 0, 0 ≤ x ≤ 1.

Again this problem will have a a unique solution for any b ∈ L2(0,∞). Rewriting(9.24) as∫ ∞

0

u′v′dx+∫ ∞

0

uv′dx+∫ ∞

0

uvdx ≥∫ ∞

0

bvdx, ∀v ∈ K. (9.25)


we may conclude that∫ ∞

0

u′v′dx+∫ ∞

0

uv′dx+∫ ∞

0

uvdx =∫ ∞

0

bvdx, ∀v ∈ C∞0 (0,∞), (9.26)

withv(x) = 0, 0 ≤ x ≤ 1

(note that, for such v, both v and −v belong to K). This implies∫ ∞

1

u′v′dx+∫ ∞

1

uv′dx+∫ ∞

1

uvdx =∫ ∞

1

bvdx, ∀v ∈ C∞0 (1,∞), (9.27)

and, as above, we conclude that u is a solution of

−∂2u− ∂u+ u = b, 1 ≤ x ≤ ∞.

The latter equation will also be satisfied at those points x ∈ (0, 1), for whichu(x) > 0. Combining these statements, we conclude that u ∈ K solves

−∂2u− ∂u+ u = b, 1 ≤ x ≤ ∞u(−∂2u− ∂u+ u− b) = 0, 0 ≤ x ≤ 1.

9.7. Elliptic boundary value problems. Let Ω be a bounded open set (withsmooth boundary) in RN , let aijNi,j=1 ⊂ L∞(Ω) satisfy∑

i,j

aij(x)ξiξj ≥ c0|ξ|2, ∀ξ ∈ RN , ∀x ∈ Ω, c0 > 0 a constant,

where | · | is a norm in RN . Let E := H10 (Ω) with

‖u‖2 = ‖u‖2H10 (Ω) =

∫Ω

|∇u|2dx,

(this is a norm, equivalent to the H1 norm, as follows from an inequality due toPoincare, see [1]), and let a(u, v) be given by

a(u, v) =∑i,j

∫Ω

aij(x)∂iu∂jvdx =∫

Ω

A∇u · ∇vdx,

where A is the N ×N matrix whose ij entry is aij and ∂iu, i = 1, . . . , N are thepartial distributional derivatives of u and ∇ is the distributional gradient. Then

|a(u, v)| ≤ c1‖v‖‖u‖, c1 = maxij

‖aij‖L∞(Ω),

|a(u, u)| ≥ c0‖u‖2.

For b ∈ L2(Ω) ⊂ H10 (Ω)∗ we obtain the existence of a unique u ∈ H1

0 (Ω) such that

a(u, v − u) ≥∫

Ω

b(v − u), ∀v ∈ H10 (Ω),

hence,

a(u, v) =∫

Ω

bvdx, ∀v ∈ H10 (Ω).

In particular, this will hold for all v ∈ C∞0 (Ω), and therefore the partial differentialequation

−∑i,j

∂j(aij∂iu) = b


has a unique solution u ∈ H10 (Ω), in the sense of distributions (see also [14], [29],

[34], [50]). As a special case we obtain that the partial differential equation

−∆u = b, (9.28)

has, in the sense of distributions, a unique solution u ∈ H10 (Ω) for every b ∈ L2(Ω)

and ∫Ω

|∇u|2dx ≤∫

Ω

b2dx. (9.29)

If Ω is a not necessarily bounded open set, one may, using arguments like theabove establish the unique solvability in H1

0 (Ω) of the elliptic equation

−∆u+ u = b, (9.30)

for every b ∈ L2(Ω) and ∫Ω

|∇u|2dx+∫

Ω

u2dx ≤∫

Ω

b2dx. (9.31)

For additional and more detailed examples see the following: [13, 18, 19, 26, 33,44, 47, 50].

10. Semilinear elliptic equations

In this chapter we shall discuss how the contraction mapping principle may beused to deduce the existence of solutions of Dirichlet problems for semilinear ellipticpartial differential equations. Such results have their origin in work of Picard [59],Lettenmeyer [49]. Our derivation is based on work contained in [35], and [54], usingthe results established in Chapter 9.

10.1. The boundary value problem. Let Ω be a bounded open set (with smoothboundary) in RN , and let

f : Ω× R× RN → Rbe a continuous function which satisfies the Lipschitz condition

|f(x, u1, v1)− f(x, u2, v2)| ≤ L1|u1 − u2|+ L2|v1 − v2|, (10.1)

for all (x, u1, v1), (x, u2, v2) ∈ Ω × R × RN , here | · | denotes both absolute valueand the Euclidean norm in RN . We shall also assume that f(·, 0, 0) ∈ L2(Ω).

We consider the following boundary value problem (in the sense of distributions)

−∆u = f(x, u,∇u), u ∈ H10 (Ω) =: H. (10.2)

We note that if u ∈ H1(Ω), then

|f(x, u,∇u)| ≤ |f(x, 0, 0)|+ L1|u|+ L2|∇u|.

Hence f may be thought of as a mapping

f : H1(Ω) → L2(Ω),

which, because of the Lipschitz condition (10.1) is, in fact, a continuous mapping.Let us denote by T , the solution operator

T : L2(Ω) → H10 (Ω)

T (w) = u,(10.3)


where u ∈ H10 (Ω) solves −∆u = w (see Chapter 9). Inequality (9.31) implies

‖u‖2H =∫

Ω

|∇u|2dx ≤∫

Ω

w2dx = ‖w‖2L2 . (10.4)

We, therefore, find that problem (10.2) is equivalent to the fixed point problem inL2(Ω)

w = f(·, T (w),∇T (w)). (10.5)

Define the operatorS : L2(Ω) → L2(Ω)

byS(w) = f(·, T (w),∇T (w)). (10.6)

Using the Lipschitz condition imposed on f , we find

|S(w1)(x)− S(w2(x)| ≤ L1|T (w1)(x)− T (w2(x)|+ L2|∇T (w1)(x)−∇T (w2(x)|

and thus

‖S(w1)− S(w2‖L2 ≤ L1‖T (w1)− T (w2‖L2 + L2‖|∇T (w1)−∇T (w2)|‖L2 (10.7)

We now recall the Poincare inequality for H10 (Ω) (see [29])

‖u‖L2 ≤ 1λ1‖u‖H , ∀u ∈ H1

0 (Ω),

where λ21 is the smallest eigenvalue of −∆ on H1

0 (Ω) (see [1], [34]). Inequalities(10.7) and (10.4) imply

‖T (w1)− T (w2)‖L2 = ‖T (w1 − w2)‖L2

≤ 1λ1‖T (w1 − w2)‖H

≤ 1λ1‖w1 − w2‖L2 .

(10.8)

We next use Green’s identity (see [34]) to compute

‖|∇T (w1)−∇T (w2)|‖2L2 = −〈w1 − w2, T (w1)− T (w2)〉≤ ‖w1 − w2‖L2‖T (w1 − w2)‖L2

≤ 1λ1‖w1 − w2‖2L2 ,

(10.9)

where 〈·, ·〉 is the L2 inner product. Combining (10.8) and (10.9) we obtain

‖S(w1)− S(w2‖L2 ≤(L1

λ1+

L2√λ1

)‖w1 − w2‖L2 . (10.10)

The operator S : L2(Ω) → L2(Ω), therefore, has a unique fixed point provided that

L1

λ1+

L2√λ1

< 1. (10.11)

As observed earlier, this fixed point is a solution of (10.5) and setting u = T (w) weobtain the solution of (10.2).

We summarize the above in the following theorem.


Theorem 10.1. Let Ω be a bounded open set (with smooth boundary) in RN , andlet

f : Ω× R× RN → Rbe a continuous function which satisfies the Lipschitz condition

|f(x, u1, v1)− f(x, u2, v2)| ≤ L1|u1 − u2|+ L2|v1 − v2|,for all (x, u1, v1), (x, u2, v2) ∈ Ω× R× RN , further assume that f(·, 0, 0) ∈ L2(Ω).Then the boundary value problem

−∆u = f(x, u,∇u), u ∈ H10 (Ω).

has a unique solution provided thatL1

λ1+

L2√λ1

< 1,

where λ21 is the principal eigenvalue of −∆ with respect to H1

0 (Ω).

In case f is independent of ∇u such a result, under the assumption that L1 besufficiently small, was already established for the two-dimensional case by Picard[59].

10.2. A particular case. In this section we shall consider problem (10.2) in thecase of one space dimension, N = 1. We shall study this problem for the case

Ω = (0, π).

The case of an arbitrary finite interval (a, b) may easily be deduced for this one.In this case λ1 = 1 and condition (10.11) becomes

L1 + L2 < 1. (10.12)

Given the boundary value problem

u′′ = f(x, u, u′), u ∈ H10 (0, π), (10.13)

with f satisfying the above assumptions, it is natural, also, to seek a solutionu ∈ C1[0, π] and to formulate an integral equation equivalent to (10.13) in thisspace, rather than L2(0, 1). This may be accomplished by using the Green’s functionG given by formula (8.8) of Chapter 8 (see e.g [36]). I.e., we have that problem(10.13) is equivalent to the integral equation problem

u(x) =∫ π

0

G(x, s)f(s, u(s), u′(s))ds, u ∈ E := C1[0, π]. (10.14)

As usual, we use‖u‖E := ‖u‖+ ‖u′‖,

where‖u‖ := max

[0,π]|u(x)|.

We define T : E → E by

T (u)u(x) :=∫ π

0

G(x, s)f(s, u(s), u′(s))ds, u ∈ E = C1[0, π].

An easy computation shows that

‖T (u1)− T (u2‖E ≤π2L1

8‖u1 − u2‖+

πL2

2‖u′1 − u′2‖,


and therefore

‖T (u1)− T (u2‖E ≤(π2L1

8+πL2

2

)‖u1 − u2‖E , (10.15)

i.e., T is a contraction mapping, whenever

π2L1

8+πL2

2< 1. (10.16)

Clearly the requirement (10.16) is different from the requirement (10.12). Thus, fora given problem, several different metric space settings may be possible, with thedifferent settings yielding different requirements.

(The condition (10.16) was already derived by Picard [59] and later by Letten-meyer [49]; many different types of requirements using different approaches may befound, e.g., in [20] and [35].)

We remark that in the above considerations we could equally have assumed thatf is a mapping

f : (0, 1)× E × E → E,

where E is a Banach space.

10.3. Monotone solutions. In this section we shall discuss some recent work in[24] concerning the nonlinear second order equation

u′′ + F (t, u) = 0, t ∈ [0,∞), (10.17)

where F : [0,∞)×[0,∞) → [0,∞) is continuous and satisfies the Lipschitz condition

|F (t, u)− F (t, v)| ≤ k(t)|u− v|, (10.18)

withk : [0,∞) → [0,∞)

a continuous function, satisfying ∫ ∞

0

tk(t)dt < 1. (10.19)

We have the following theorem on monotone solutions of (10.17).

Theorem 10.2. Let the above conditions hold and assume that there exists M > 0such that for any u ∈ X, where

X := u ∈ C[0,∞) : 0 ≤ u(t) ≤M, t ∈ [0,∞),

we have ∫ ∞

0

tF (t, u(t))dt ≤M. (10.20)

Then (10.17) has a monotone solution u : [0,∞) → [0,M ] such that

limt→∞

u(t) = M.

Proof. LetE := u ∈ C[0,∞) : ‖u‖ := sup

t∈[0,∞)

|u(t)| <∞,

then E is a Banach space and X is a closed subset of E and hence a completemetric space with respect to the metric defined by the norm in E.


Next consider the mapping T on X defined by

(Tu)(t) := M −∫ ∞

t

(τ − t)F (τ, u(τ))dτ. (10.21)

Then, since for u ∈ X

0 ≤∫ ∞

t

(τ − t)F (τ, u(τ))dτ ≤∫ ∞

0

τF (τ, u(τ))dτ ≤M, (10.22)

it follows that T : X → X.On the other hand, for u, v ∈ X we have

|(Tu)(t)− T (v)(t)| ≤∫ ∞

t

(τ − t)|F (τ, u(τ))− F (τ, v(τ))|dτ

≤∫ ∞

t

(τ − t)k(τ)|u(τ)− v(τ)|dτ

≤∫ ∞

0

τk(τ)dτ‖u− v‖.

Hence T is a contraction onX and therefore has a unique fixed point. If u ∈ X is thefixed point of T , then it easily follows that u is monotone and limt→∞ u(t) = M .

For applications of this result to nonoscillation theory of second order differentialequations, see [24]

11. A mapping theorem in Hilbert space

In this chapter we shall discuss a result of Mawhin [53] on nonlinear mappings inHilbert spaces which has, among others, several interesting applications to existencequestions about periodic solutions of nonlinear conservative systems (see [53] and[74] for many references to this interesting problem area; see also [3] and [67] forfurther directions).

11.1. A mapping theorem. Let H be a real Hilbert space with scalar product〈·, ·〉 and norm ‖ · ‖. We assume that

L : domL ⊂ H → H,

is a linear, self-adjoint operator (domL is the domain of L) and

N : H → H

is a nonlinear mapping which is Frechet differentiable there, with symmetric Frechetderivative N ′ (see [63], [66]).

For a given linear operator

A : domA ⊂ H → H,

we denote by ρ(A), σ(A), r(A), respectively the resolvent set, spectrum, and thespectral radius of the operator A (see [43]). Also one writes, for a given linearoperator A,

A ≥ 0, if, and only if, 〈Au, u〉 ≥ 0, ∀u ∈ domA.and

A ≥ B, if, and only if, A−B ≥ 0,(here, of course, A−B is an operator defined on the intersection of the domains ofA and B).

We establish the following surjectivity theorem.


Theorem 11.1. Suppose L and N are as above and there exist real numbers λ, µ,p, q such that

λ < q ≤ p < µ, [λ, µ] ⊂ ρ(L),and that

qI ≤ N ′(u) ≤ pI, ∀u ∈ H,where I : H → H is the identity mapping. Then

L−N : domL→ H

is a bijection.

Proof. For any ν ∈ (λ, µ), and v ∈ H the equation

Lu−N(u) = v,

is equivalent to the equation

(L− νI)u− (N − νI)(u) = v,

orAu−B(u) = v, (11.1)

whereA := L− νI, B := N − νI.

It follows from the assumptions that B has the symmetric Frechet derivative givenby

B′(u) = N ′(u)− νI.

Since ν ∈ ρ(L), it follows that

A−1 = (L− νI)−1

exists and is a bounded operator and further that

[λ− ν, µ− ν] ⊂ ρ(A).

Hence, since A is self-adjoint,

σ(A) ⊂ (−∞, λ− ν) ∪ (µ− ν,∞).

One may further deduce that

σ(A−1) ⊂((λ− ν)−1, (µ− ν)−1

).

Hence‖A−1‖ = r(A−1) ≤ max(ν − λ)−1, (µ− ν)−1 =: α. (11.2)

All of the above follow from properties of linear operators (see [43]). Next, we notethat

‖B(u)−B(v)‖ ≤ supτ∈[0,1]

‖B′(u+ τ(v − u)‖‖u− v‖

≤ supw∈H

‖N ′(w)− νI‖‖u− v‖.

On the other hand

(q − ν)I ≤ N ′(u)− νI = B′(u) ≤ (p− ν)I,

and hence for each u ∈ H,

〈(q − ν)Iv, v〉 ≤ 〈B′(u)v, v〉 ≤ 〈(p− ν)Iv, v〉,∀v ∈ H,


or

(q − ν)‖v‖2 ≤ 〈B′(u)v, v〉 ≤ (p− ν)‖v‖2.

The latter implies (see again [43])

‖B′(u)‖ ≤ max(|q − ν|, |p− ν|) =: β. (11.3)

Now, equation (11.1) is equivalent to the equation

u = A−1(B(u) + v). (11.4)

We next note that

‖A−1(B(u) + v)−A−1(B(w) + v)‖ ≤ ‖A−1‖‖B(u)−B(w)‖≤ αβ‖u− w‖

(11.5)

(see (11.2), (11.3)), and hence, for every v, the mapping

u 7→ A−1(B(u) + v)

will be a contraction mapping provided αβ < 1, will be satisfied, whenever

p+ λ

2< ν <

q + µ

2.

The latter will hold, for example if we choose

ν =p+ q

2or ν =

λ+ µ

2.

We remark that it follows from the results in [43] that

‖A−1‖ =1

dist(ν, σ(L))

and hence (11.1) may be rewritten as

‖A−1(B(u) + v)−A−1(B(w) + v)‖ ≤ ‖A−1‖‖N(u)−N(v)− ν(u− v)‖. (11.6)

Using this, one obtains the following more general version of Theorem 11.1.

Theorem 11.2. Suppose L and N are as above and there exists a real numberν ∈ ρ(L), such that 0 < dist(ν, σ(L)) and

‖N(u)−N(v)− ν(u− v)‖ ≤ k‖u− v‖, ∀u, v ∈ H,

where

k < dist(ν, σ(L)).

Then L−N : domL→ H is a bijection.


11.2. Periodic solutions of conservative systems. In this section we shall dis-cuss the existence of 2π-periodic solutions of the system of second order differentialequations

u′′ + n(u) = h(t), (11.7)

where n : RN → RN is a C1 function such that its Frechet derivative n′(u) satisfies

rI ≤ n′(u) ≤ sI,

where, for some m,

m2 < r ≤ s < (m+ 1)2, m ∈ 0, 1, 2, . . . ,

r and s are given andh : (−∞,∞) → RN

is a 2π− periodic function with h ∈ L2(0, 2π).The following result (for more general versions; see, e.g. [53] or [74]) is valid:

Theorem 11.3. Let the above assumptions hold. Then there exists a unique 2π−periodic solution of (11.7).

Proof. We let H = L2(0, 2π) with inner product

〈u, v〉 :=12π

∫ 2π

0

uvdx.

LetdomL = u ∈ H : u, u′ absolutely continuous, periodic, u′′ ∈ H

L : domL→ H, u 7→ u′′.

Then L is self-adjoint and it follows, using elementary ordinary differential equationsresults, that

σ(L) = −n2, n = 0, 1, . . . .It is also an easy exercise to verify that, if we define by

Nu := −n(u(·)),

then N : H → H, and(N ′(u))v(t) = −n′(u(t))v(t).

Using the other assumptions, one sees that all hypotheses of Theorem 1 hold,proving the theorem.

12. The theorem of Cauchy-Kowalevsky

In this chapter we provide one of the fundamental theorems of the theory of par-tial differential equations, the existence theorem of Cauchy-Kowalevsky. It marksthe starting point for the general theory of partial differential equations, both fromthe point of view of analysis and as well as from the historical point of view. Itbasically asserts that the noncharacteristic Cauchy problem with holomorphic coef-ficients and holomorphic initial data is well-posed. The proof is based on an elegantpaper of Walter [73] and follows the treatment provided in [68].


12.1. The setting. Let C denote the set of complex numbers and Cn denote then-dimensional space of n-tuples of complex numbers z = (z1, . . . zn). For z ∈ Cn,we define the norm of z by |z| = max1≤j≤n |zj |.Definition 12.1. Let G be a domain in Cn. A function f : G→ C will be said tobe holomorphic in G, if f and ∂f/∂zj = fzj

, j = 1, . . . n, are continuous in G.

Let Ω be an open set in Cn such that the boundary Γ = ∂Ω is nonempty. Let

d(z) := dist(z,Γ)

denote the distance from z to the boundary of Ω,

dist(z,Γ) := infζ∈Γ

|z − ζ|.

Let η be a positive real number and define the set Ωη by

Ωη := (t, z) : z ∈ Ω, |t| < ηd(z),and either t ∈ R (real case) or t ∈ C (complex case). In geometrical terms in thereal case, i.e., Ωη is the double cone, with base Ω whose sides have slope η. Wedenote by Ωt ⊂ Cn the set

Ωt = z ∈ Ω : (t, z) ∈ Ωη,and by Γt the boundary of Ωt. If z ∈ Ωt, then d(z) > |t|/η. Geometrically, in thereal case, the set Ωt is the projection onto Cn of the base of that part of Ω whichlies above t (t > 0) or below t (t < 0). For z ∈ Ωt, we define d(t, z) by

d(t, z) := d(z)− |t|η. (12.1)

The function d(t, z) is positive and represents the distance from z ∈ Ωt to Γt. Thefollowing property of d(t, z) will be needed later in the proof of the theorem.

Lemma 12.2. If z′ ∈ Cn satisfies |z − z′| = r < d(t, z) for some z ∈ Ωt, thenz′ ∈ Ωt and

d(t, z′) ≥ d(t, z)− r. (12.2)

12.2. The linear case. We will first give a proof of the Cauchy-Kowalevsky The-orem in the case where the equation is linear. I.e., we consider the initial valueproblem

ut = A(t, z)u+n∑j=1

Bj(t, z)uzj+ C(t, z), z ∈ Ω,

u(0, z) = f(z) z ∈ Ω.

(12.3)

Here, z ∈ Ω, t ∈ R (real case), or t ∈ C (complex case) and u, ut, uzj, and C(t, z) are

complex valued column vectors in Cm; and A(t, z) and Bj(t, z) are complex valuedm ×m matrices. The set G := Ωη is the set defined above. If we integrate bothsides of equation (12.3) with respect to t and use the initial condition to evaluatethe integral of the left hand side, we obtain the equivalent integral formulation ofthe problem

u(t, z) = f(z) +∫ t

0

C(τ, z) dτ

+∫ t

0

[A(τ, z)u(τ, z) +

n∑j=1

Bj(τ, z)uzj(τ, z)

]dτ.

(12.4)


Remark 12.3. In the complex case (t ∈ C), the integration in equation (12.4) istaken along the straight line which connects 0 to t in C.

Definition 12.4. By a solution to equation (12.4), we mean a function u(t, z)which is continuous in G, holomorphic in z for fixed t (real case) or holomorphic in(t, z) (complex case), and which satisfies (12.4).

Under suitable assumptions on the coefficients A(t, z), Bj(t, z), C(t, z), and f(z),we shall see that if u(t, z) is a solution to one of the two problems (12.4) or (12.3),then u is of class C1 and is a solution to the other of the two problems, i.e., the twoformulations of the problem are equivalent. Throughout we shall use the operatornorm for matrices, |A| = max1≤k≤m

∑nj=1 |ajk|, induced by the vector norm | · |.

Before stating the main result of the section, we will first prove a lemma whichgives a crucial bound on the derivatives of holomorphic functions. The result isusually referred to as Nagumo’s lemma.

Lemma 12.5. Let Ω be a domain in Cn and let f : Ω → Cm be holomorphic andlet p ≥ 0. If, for z ∈ Ω,

|f(z)| ≤ c

dp(z),

then|fzj

(z)| ≤ Cpc

dp+1(z),

whereCp = (1 + p)

(1 +

1p

)p< e(p+ 1), C0 = 1.

Proof. Consider first the case of a single function of a single complex variable. Letψ : C → C be holomorphic in the disk ζ ′ : |ζ − ζ ′| ≤ r. Then by the Cauchyintegral formula, see [63]

ψ′(ζ) =1

2πi

∮|ζ−ζ′|=r

ψ(ζ ′)(ζ ′ − ζ)2

dζ ′.

Thus, we obtain

|ψ′(ζ)| = 12π

∣∣∣ ∮|ζ−ζ′|=r

ψ(ζ ′)(ζ ′ − ζ)2

dζ ′∣∣∣

≤ 12π

∮|ζ−ζ′|=r

|ψ(ζ ′)||ζ ′ − ζ|2

|dζ ′|

≤ 12πr2

max|ζ−ζ′|=r

|ψ(ζ ′)| · (2πr)

=1r

max|ζ−ζ′|=r

|ψ(ζ ′)|.

Now let f : Ω → Cm be holomorphic. We apply the result that we have justobtained to f(z), with ζ = zj . If 0 < r < d(z), then

|fzj(z)| ≤ 1

rmax

|z−z′|=r|f(z′)|

≤ c

rmax

|z−z′|=r

1dp(z′)

≤ c

r(d(z)− r)p.


Where d(z′) ≥ d(z)− r because of Lemma 12.2. The choice r = d(z)/(p+ 1) givesthe (optimal) value stated in the lemma.

We next establish the linear Cauchy-Kowalevsky theorem.

Theorem 12.6. Let the functions A(t, z), Bj(t, z) and C(t, z) be continuous in(t, z) and holomorphic in z for fixed t and let f(z) be holomorphic in z. Supposethat there exist constants α, βj, γ, δ and p such that for every (t, z) in G,

|A(t, z)| ≤ α

d(t, z), |Bj(t, z)| ≤ βj ,

|C(t, z)| ≤ γ

dp+1(t, z), |f(z)| ≤ δ

dp(z).

(12.5)

Suppose further that η > 0 is such that

α

p+

(1 +

1p

)p+1 n∑j=1

βj <1η. (12.6)

Then (12.4) has a unique solution u(t, z) in G and for some constant c this solutionsatisfies

|u(t, z)| ≤ c

dp(t, z).

Note that condition (12.6) is a smallness condition on η and thus makes thetheorem a local one, i.e., guarantees the existence of solutions for small time. Thecondition, as will be seen, is imposed in order for the contraction mapping principleto be applicable.

Proof. Let E be the normed linear space defined by

E := u ∈ C0(G,Cm) : u is holomorphic in z and ‖u‖ <∞,

where the norm on E is defined by

‖u‖ := supGdp(t, z)|u(t, z)|.

Note that convergence in this norm implies uniform convergence on compact subsetsof G; hence limit functions are holomorphic in z and E is complete.

We write equation (12.4) in the form

u = g + Tu,

where g is given by the equation

g(t, z) := f(z) +∫ t

0

C(τ, z) dτ,

and T is the linear operator given by

(Tu)(t, z) :=∫ t

0

[A(τ, z)u(τ, z) +

n∑j=1

Bj(τ, z)uzj(τ, z)

]dτ.

We first show that g + Tu ∈ E if u ∈ E. Consider each of the terms in turn. Forthe function f(z) we have, using (12.5),

dp(t, z)|f(z)| ≤ dp(t, z)δ

dp(z)≤ δ.


The last inequality follows from the fact that d(z) ≥ d(z)− |t|/η = d(t, z). Takingthe supremum of the left-hand side of this inequality, we get ‖f‖ ≤ δ.

Before estimating the next term in g, we note that a direct integration gives∣∣∣ ∫ t

0

dτ

dp+1(τz)

∣∣∣ ≤ ∫ |t|

0

ds(d(z)− s

η

)p+1 ≤η

pdp(t, z). (12.7)

Therefore, for the second term in g, we have∣∣∣ ∫ t

0

C(τ, z) dτ∣∣∣ ≤ ∣∣∣ ∫ t

0

|C(τ, z)| dτ∣∣∣

≤ γ∣∣∣ ∫ t

0

dτ

dp+1(τ, z)

∣∣∣<

γη

pdp(t, z).

Again, we have used the bound from (12.5). Multiplying both sides of this inequalityby dp(t, z) and taking the supremum over G, we obtain∥∥∫ t

0

C(τ, z) dτ∥∥ ≤ γη

p.

It follows from this last inequality and the bound on f , that g ∈ E.According to the definition of the norm on E, we have the inequality

|u(t, z)| ≤ ‖u‖dp(t, z)

.

Starting from this inequality, we may now apply Nagumo’s lemma, Lemma 12.5, tothe region Ωt, using the distance function d(t, z). As a result, we get the estimate

|uzj (t, z)| ≤ Cp‖u‖

dp+1(t, z).

Combining (12.5) with the bounds just obtained on u and uzj, we get the estimates

|A(t, z)u(t, z)| ≤ α‖u‖dp+1(t, z)

,

|Bj(t, z)uzj(t, z)| ≤ ‖u‖

dp+1(t, z)βjCp.

Hence, with β :=∑nj=1 βj and using the estimate (12.7), we have

|(Tu)(t, z)| ≤ ‖u‖(α+ βCp)∣∣∣ ∫ t

0

dτ

dp+1(τ, z)

∣∣∣≤ 1p(α+ βCp)η

‖u‖dp(t, z)

.

If we multiply both sides of this inequality by dp(t, z) and take the supremum ofthe left hand side, we get the final estimate

‖Tu‖ ≤ q‖u‖, (12.8)

where

q =(αp

+ β(1 +

1p

)p+1)η.


This shows that Tu ∈ E. Furthermore, the constant q satisfies q < 1 by thehypotheses of the theorem, hence, the contraction mapping principle may be appliedto obtain a unique solution of the equation u = g + Tu.

Example 12.7. Consider the equation

ut = buz,

(here n = 1, b =constant) subject to the initial condition

u(0, z) = φ(z).

In this example α = 0 and β = |b|. The solution of this initial value problem isgiven by u(t, z) = φ(bt + z). Suppose that Ω is the disk |z| < 1 and that φ isholomorphic in Ω. Then the solution exists for |bt + z| < 1, and if φ cannot becontinued analytically beyond the unit circle, then this is best possible. If we varyb, but keep |b| = β fixed, then the largest region common to all those regions ofexistence is the circular cone β|t| < 1− |z|, i.e., η = 1/β is best possible.

An added advantage to the approach that has been adopted for the Cauchy-Kowalevsky Theorem, is that the fixed point of a contractive mapping S is notonly unique, it also depends continuously on the operator S. Thus we can easilyobtain results on continuous dependence of the solution on the coefficients and theinitial data.

12.3. The quasilinear case. We are now prepared to present the result for thegeneral case. It may be shown that the Cauchy problem for a general nonlinearsystem with holomorphic coefficients can be reduced, by a change of variables, toan equivalent initial value problem for a first order system which is quasilinear.Therefore, it is sufficient to state and prove the result for the initial value problemfor a first order quasilinear system. Furthermore, if u satisfies an initial condition ofthe form u(0, z) = f(z), then the substitution of the function u(t, z) = f(z)+v(t, z)gives a new differential equation for v which is again a first order quasi-linearequation of the same type and satisfies the initial condition v(t, z) = 0. Takingthese observations into account, we shall consider the following initial value problem

ut =n∑j=1

Bj(t, z, u)uzj + C(t, z, u), (t, z) ∈ G,

u(0, z) = 0, z ∈ Ω.

(12.9)

As was the case for linear equations, the Bj are m×m matrices and C is a columnvector of length m. The set G = Ωη is again defined by the inequality |t| < ηd(z),where η is defined later in the theorem and d(z) is an appropriately determined“distance” function. The set BR := BR(0) is the open ball z : |z| < R in Cn. Weshall assume that d(z) is bounded on Ω and satisfies the inequalities

0 < d(z) ≤ dist(z,Γ), |d(z)− d(z′)| ≤ |z − z′|, z, z′ ∈ Ω.

If G, Ωt, and d(t, z) are defined as before, using the new function d(z), thenLemma 12.2 remains valid. Moreover, 0 < d(t, z) < dist(z,Γt), for z ∈ Ωt. Asa consequence, Lemma 12.5 remains true. Although the region G of existence ob-tained is less than optimal under this assumption, we derive the benefit that theproof of the theorem is greatly simplified. We now state and prove the Cauchy-Kowalevsky theorem for the quasilinear case.


Theorem 12.8. Let the functions Bj(t, z, u) and C(t, z, u) be continuous in Ωη ×BR and holomorphic with respect to z and u (real case), or holomorphic in Ωη×BR(complex case). Suppose further, that the following estimates hold on Ωη ×BR:

|Bj(t, z, u)| ≤ βj ,√d(t, z)|Bj(t, z, u)−Bj(t, zv)| ≤ β′j |u− v|, j = 1, . . . , n,

|C(t, z, u)| ≤ γ√d(t, z)

, d(t, z)|C(t, z, u)− C(t, z, v)| ≤ γ′|u− v|.(12.10)

If η > 0 is such that

2η√d0(β + γ) < R, η(3

√3(β + γ) + 2β) ≤ 1, ηeβ′ < 1,

where d0 = sup d(z) <∞, β =∑nj=1 βj, and β′ =

∑nj=1 β

′j. Then the initial value

problem (12.9) has a unique solution which exists in Ωη.

Proof. The proof proceeds along lines similar to the linear case. We consider anequivalent integral formulation of (12.9) given by

u(t, z) =∫ t

0

[ n∑j=1

Bj(τ, z, u(τ, z))uzj(τ, z) + C(τ, z, u(τ, z))

]dτ (12.11)

and treat it as a fixed point equation of the form u = Su to which the contractionmapping principle, can be applied. As the underlying Banach space E, we use thesame space as in the proof of Theorem 12.6. However, in contrast to the linearcase, the operator S is not globally defined on E and it will be necessary to definea proper closed subset F of E which is mapped into itself by S.

Let F be the closed subset of E defined by

F :=u ∈ E : |u(t, z)| ≤ ρ, |uzj (t, z)| ≤ 1/

√d(t, z)

,

where ρ := 2η√d0(β+ γ) < R. Let u ∈ F and set v = Su. Then, using the bounds

in (12.10), we obtain

|vt(t, z)| =∣∣ n∑j=1

Bj(t, z, u(t, z))uzj(t, z) + C(t, z, u(t, z))

∣∣≤

n∑j=1

|Bj(t, z, u(t, z))||uzj(t, z)|+ |C(t, z, u(t, z))|

≤ β + γ√d(t, z)

.

From this inequality and the fundamental theorem of calculus, it follows that

|v(t, z)| =∣∣∣ ∫ t

0

vt(τ, z) dτ∣∣∣ ≤ (β + γ)

∫ |t|

0

ds√d(s, z)

.


A direct integration then yields∫ |t|

0

ds√d(s, z)

=∫ |t|

0

ds√d(z)− s

η

= −2η√d(z)− s

η

∣∣∣|t|0

≤ 2η√d(z)

≤ 2η√d0.

Therefore,|v(t, z)| ≤ 2η(β + γ)

√d0 = ρ,

which is the first inequality in the definition of the set F . In order to verify thesecond inequality in the definition of F , we estimate the derivatives of v. Using theinequalities in (12.10) and applying Nagumo’s Lemma, Lemma 12.5, we obtain thebounds ∣∣ ∂

∂zkC(t, z, u(t, z))

∣∣ ≤ γC1/2

d3/2(t, z)∣∣ ∂

∂zkBj(t, z, u(t, z))

∣∣ ≤ βjd3/2(t, z)

,

and from the second inequality in the definition of F and Nagumo’s Lemma,

|uzjzk| ≤

C1/2

d(t, z).

Here C1/2 = 3√

3/2. The product formula for derivatives, leads to the the inequality

|vt,zk| =

∣∣∣ n∑j=1

[(Bj)zkuzj

+Bjuzjzk] + Czk

∣∣∣≤

n∑j=1

|(Bj)zk||uzj |+ |Bj ||uzjzk

|+ |Czk|

≤(β + γ)C1/2 + β

d3/2(t, z).

A direct integration with respect to s establishes the inequality∫ |t|

0

ds

d3/2(s, z)≤ 2η√

d(t, z).

Therefore, using the fundamental theorem of calculus, we have the estimate

|vzj | ≤∣∣ ∫ t

0

vt,zj (τ, z) dτ∣∣

≤ ((β + γ)C1/2 + β)∫ |t|

0

1(d3/2(τ, z))

dτ

≤ 1√d(t, z)

.


This last inequality verifies that v satisfies the second inequality in the definitionof F , as well. It follows from the above that Su ∈ F whenever u ∈ F , i.e., S mapsF into itself.

To complete the proof, we need to show that S is a contraction mapping. Let uand v be in F and consider the t derivative of the difference

(Su− Sv)t =n∑j=1

[Bj(t, z, u)−Bj(t, z, v)]uzj

+n∑j=1

Bj(t, z, v)[uzj− vzj

] + C(t, z, u)− C(t, z, v).

The definition of the Banach space E gives us the inequality

|u− v| ≤ ‖u− v‖dp(t, z)

.

Applying Nagumo’s lemma to this inequality, gives the estimate

|uzj− vzj

| ≤ Cp‖u− v‖dp+1(t, z)

.

Combining these results with the hypotheses (12.10), we obtain the following in-equality for the difference in the derivatives

|(Su− Sv)t| ≤( n∑j=1

β′j

) |u− v|d(t, z)

+n∑j=1

βj |uzj − vzj |+ γ′|u− v|d(t, z)

≤(β′ + βCp + γ′

) ‖u− v‖dp+1(t, z)

.

Thus, making use of (12.7), and the fundamental theorem of calculus, we get

|Su− Sv| ≤∣∣ ∫ t

0

(Su− Sv)t dt∣∣ ≤ η

p(β′ + Cpβ + γ′)

‖u− v‖dp(t, z)

.

Hence, when we multiply by dp(t, z) and then take the supremum over Ωη, we get

‖Su− Sv‖ ≤ η

p(β′ + Cpβ + γ′)‖u− v‖.

It follows from our hypotheses on η that S is a contraction mapping for p sufficientlylarge and S has a unique fixed point u ∈ F which is the solution to (12.11).

An example to which the above, for example may be applied, is the the initialvalue problem for a Burger’s type equation

ut = −(1 + z2 + u)uz − 2z(1 + z2 + u), z ∈ Ω, 0 < t

u(0, z) = 0, z ∈ Ω,

where Ω = z : |z| < 1 is the unit disk, centered at the origin. In the terms ofthe notation of Theorem 12.8, we have B(t, z, u) = −(1 + z2 + u) and C(t, z, u) =−2z(1 + z2 + u), which are entire functions. Note also that B(t, z, u) and C(t, z, u)are real valued for real values of t, z, and u. The distance function d(z) is given by


d(z) = 1 − |z|, d0 = 1, and d(t, z) = 1 − |z| − |t|η. Let R > 0 be given. Then theconditions (12.10) are satisfied with the constants

β = 2 +R, β′ = 1,

γ = 2(2 +R), γ′ = 2.

If η is chosen such that

η ≤ min R

6(2 + r),

1(9√

3 + 2)(2 +R),

1e

,

then the hypotheses of Theorem 12.8 are satisfied and the existence of a uniqueholomorphic solution defined on Ωη follows.

References

[1] R. Adams, Sobolev Spaces, Academic Press, New York, 1975.

[2] E. L. Allgower and K. Georg, Numerical Continuation Methods, An Introduction, Springer-Verlag, New York, 1990.

[3] H. Amann, On the unique solvability of semilinear operator equations in Hilbert spaces, J.

Math. Pures Appl., 61 (1982), pp. 149–175.[4] L. Amerio and G. Prouse, Almost-periodic Functions and Functional Equations, vol. VIII of

The University Series in Higher Mathematics, Van Nostrand, New York, 1971.[5] S. Banach, Sur les operations dans les ensembles abstraits et leur applications aux equations

integrales, Fundamenta Math., 3 (1922), pp. 133–181.

[6] M. Barnsley, Fractals Everywhere, Academic Press, San Diego, 1988.[7] F. Bauer, An elementary proof of the Hopf inequality for positive operators, Num. Math., 7

(1965), pp. 331–337.

[8] J. Bebernes and D. Eberly, Mathematical Problems from Combustion Theory, vol. 83 ofApplied Math. Sci., Springer Verlag, New York, 1989.

[9] C. Bessaga, On the converse of the Banach fixed - point principle, Colloq. Math., 7 (1959),

pp. 41–43.[10] G. Birkhoff, Extensions of Jentzsch’s theorem, Trans. Amer. Math. Soc., 85 (1957), pp. 219–

227.

[11] G. Birkhoff and G. Rota, Ordinary Differential Equations, Blaisdell, Waltham, 1969.[12] H. Bohr, Fastperiodische Funktionen, Springer, Berlin, 1932.

[13] H. Brezis, Problemes unilateraux, J. Math. Pures Appl., 51 (1972), pp. 1–168.[14] H. Brezis, Analyse Fonctionelle, Masson, Paris, 1983.

[15] A. Bruckner, J. Bruckner, and B. Thomson, Real Analysis, Prentice Hall, Upper Saddle River,

1997.[16] P. Bushell, Hilbert’s metric and positive contraction mappings in a Banach space, Arch. Rat.

Mech. Anal., 52 (1973), pp. 330–338.

[17] R. Caccioppoli, Un teorema generale sull’esistenza de elemente uniti in una transformazionefunzionale, Rend. Acad. Naz. Linzei, 11 (1930), pp. 31–49.

[18] M. Chipot, Variational Inequalities and Flow in Porous Media, vol. 52 of Applied Math.

Sciences, Springer, New York, 1984.

[19] P. G. Ciarlet and P. Rabier, Les Equations de von Karman, Springer, Berlin, 1980.[20] W. Coles and T. Sherman, Convergence of successive approximations for nonlinear two point

boundary value problems, SIAM J. Appl. Math., 15 (1967), pp. 426–454.[21] A. Coppel, Almost periodic properties of ordinary differential equations, Ann. Mat. Pura

Appl., 76 (1967), pp. 27–50.[22] C. Corduneanu, Almost Periodic Functions, John Wiley and Sons, New York, 1968.

[23] K. Deimling, Nonlinear Functional Analysis, Springer, 1985.[24] S. Dube and A. Mingarelli, Note on a non-oscillation theorem of Atkinson, Electr. J. Differ-

ential Equations, 2004 (2004), pp. 1–6.[25] J. Dugundji, Topology, Allyn and Bacon, Boston, 1973.

[26] G. Duvaut and J. L. Lions, Les Inequations en Mecanique et en Physique, Dunod, Paris,1972.


[27] M. Edelstein, An extension of Banach’s contraction principle, Proc. Amer. Math. Soc., 12

(1961), pp. 7–10.

[28] M. Edelstein, On fixed and periodic points under contraction mappings, J. London Math.Society, 37 (1962), pp. 74–79.

[29] L. C. Evans, Partial Differential Equations, American Math. Soc., Providence, 1998.

[30] J. Favard, Lecons sur les fonctions presque-periodiques, Gauthier-Villars, Paris, 1933.[31] A. M. Fink, Almost Periodic Differential Equations, vol. 377, Springer Verlag, Berlin, 1974.

[32] P. M. Fitzpatrick, Advanced Calculus: A Course in Mathematical Analysis, PWS Publishing

Company, Boston, 1996.[33] A. Friedman, Variational Principles and Free Boundary Value Problems, Wiley-Interscience,

New York, 1983.

[34] D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of Second Order,Springer Verlag, Berlin, 1983.

[35] D. Hai and K. Schmitt, Existence and uniqueness results for nonlinear boundary value prob-lems, Rocky Mtn. J. Math., 24 (1994), pp. 77–91.

[36] P. Hartman, Ordinary Differential Equations, Birkhauser, Boston, second ed., 1982.

[37] K. Hoffman, Analysis in Euclidean Spaces, Prentice Hall, Englewood Cliffs, 1975.[38] E. Hopf, An inequality for positive linear integral operators, J. Math. Mech., 12 (1963),

pp. 683–692.

[39] E. Hopf, Remarks on my paper ”An inequality for positive linear integral operators”, J. Math.Mech., 12 (1963), pp. 889–892.

[40] J. Hutchinson, Fractals and self-similarity, Indiana Univ. J. Math., 30 (1981), pp. 713–747.

[41] J. Jachymski, A short proof of the converse of the contraction mapping principle and somerealted results, Top. Meth. Nonl. Anal., 15 (2000), pp. 179–186.

[42] H. Jeggle, Nichtlineare Funktionalanalysis, Teubner, 1979.

[43] T. Kato, Perturbation Theory for Linear Operators, Springer Verlag, Berlin, 1966.[44] D. Kinderlehrer and G. Stampacchia, An Introduction to Variational Inequalities, Acad.

Press, New York, 1980.[45] E. Kohlberg and J. W. Pratt, The contraction mapping approach to the Perron Frobenius

theory: why Hilbert’s metric, Math. Oper. Res., 7 (1982), pp. 192–210.

[46] M. A. Krasnosel’skii, Positive Solutions of Operator Equations, Noordhoff, Groningen, 1964.[47] V. Le and K. Schmitt, Global Bifurcation in Variational Inequalities: Applications to Obstacle

and Unilateral Problems, Springer, New York, 1997.

[48] S. Leon, Linear Algebra with Applications, Prentice Hall, Upper Saddle River, 1998.

[49] F. Lettenmeyer, Uber die von einem Punkt ausgehenden Integralkurven einer Differential-

gleichung zweiter Ordnung, Deutsche Math., 7 (1944), pp. 56–74.[50] J. L. Lions and E. Magenes, Non Homogeneous Boundary Value Problems and Applications,

Springer, Berlin, 1972.

[51] J. L. Lions and G. Stampacchia, Variational inequalities, Comm. Pure Appl. Math., 20(1967), pp. 493–519.

[52] J. Liouville, Sur l’equation aux derivees partielles d2 log λdudv

± 2λa2 = 0, J. Math. Pures Appl.,

18 (1853), pp. 71–72.[53] J. Mawhin, Contractive mappings and periodically perturbed conservative systems, Arch.

Math.2, Brno, 12 (1976), pp. 67–74.[54] J. Mawhin, Two point boundary value problems for nonlinear second order differential equa-

tions in Hilbert space, Tohoku Math. J., 32 (1980), pp. 225–233.[55] S. Nadler, Multivalued contraction mappings, Pac. J. Math., 30 (1969), pp. 475–488.

[56] I. P. Natanson, Theory of Functions of a Real Variable, vol. 2, Ungar, New York, 1960.[57] H. O. Peitgen, H. Jurgens, and D. Saupe, Chaos and Fractals: New Frontiers of Science,

Springer, New York, 1992.[58] M. E. Picard, Traite d’analyse, Gauthiers-Villars, Paris, 2d ed., 1893.[59] M. E. Picard, Lecons sur quelques problemes aux limites de la theorie des equations

differentielles, Gauthiers-Villars, Paris, 1930.

[60] A. Pietsch, History of Banach Spaces and Linear Operators, Birkhaeuser, 2007.[61] J. Rainermann and V. Stallbohm, Eine Anwendung des Edelsteinschen Fixpunktsatzes auf

Integralgleichungen vom Abel-Liouvillschen typus, Arch. Math., 22 (1971), pp. 642–647.[62] H. L. Royden, Real Analysis, 3rd ed., Macmillan Publishing Co., New York, 1988.[63] W. Rudin, Real and Complex Analysis, McGraw Hill, New York, 1966.


[64] T. Saaty, Modern Nonlinear Equations, Dover Publications, New York, 1981.

[65] H. Schaefer, Topological Vector Spaces, McMillan, New York, 1967.

[66] M. Schechter, Principles of Functional Analysis, Academic Press, New York, 1971.[67] K. Schmitt and H. Smith, Fredholm alternatives for nondifferentiable operators, Rocky Mtn.

J. Math., (1982), pp. 817–841.

[68] K. Schmitt and R. Thompson, Nonlinear Analysis and Differential Equations: An Introduc-tion, Lecture Notes, University of Utah, Department of Mathematics, 1998.

[69] R. E. Showalter, Hilbert Space Methods for Partial Differential Equations, Pitman, London,

1977.[70] G. Strang, Linear Algebra and its Applications, Harcourt, Brace, Jovanovich, San Diego,

3rd ed., 1988.

[71] A. Taylor, Introduction to Functional Analysis, Wiley and Sons, New York, 1972.[72] B. Thomson, A. Bruckner, and J. Bruckner, Elementary Real Analysis, Prentice Hall, Upper

Saddle River, 2001.[73] W. Walter, An elementary proof of the Cauchy-Kowalevsky theorem, Amer. Math. Monthly,

92 (1985), pp. 115–126.

[74] J. Ward, The existence of periodic solutions for nonlinearly perturbed conservative systems,Nonlinear Analysis, TMA, 3 (1979), pp. 697–705.

[75] J. Weissinger, Zur Theorie und Anwendung des Iterationsverfahrens, Math. Nachrichten, 8

(1952), pp. 193–212.[76] K. Yosida, Functional Analysis, 6th ed., Springer, New York, 1995.

Index

Symbols

L1− norm9

A

Abel-Liouville equation47

accumulation point3

almost everywhere9

almost periodic54, 55

Archimedean cone30

asymptotically stable50, 51

B

Banach fixed point theorem15

Banach lattice37

Banach space4

bijection74

bilinear form62

bounded set3

Burger’s equation84

C

Cantor set28

Cauchy integral formula78

Cauchy problem81

Cauchy sequence3

Cauchy-Kowalevsky theorem76

closed ball3

closed set3

closure3

cluster point4

coercive62

compact3

compact support8

complete3

completion7

cone30

contraction constant15

contraction correspondence22

contraction mapping15, 46

contraction mapping principle15

contraction ratio34

converse23

D

dense3

diameter3

Dirichlet problem69

distance3

distribution10

distributional derivative10, 67

distributional gradient68

dual space62

E

eigenvalue70

elliptic boundary value problem68

equivalence classes7

equivalence relation7

equivalent norm6

F

fast Cauchy sequence9

fast convergent13

Frechet differentiable57, 73

fractals28

full rank29

fundamental theorem of calculus82

G

Green’s identity70

H

Hausdorff maximal principal24

Hausdorff metric12, 25

Hilbert space5, 21

Hilbert’s projective metric30, 32

holomorphic77

homogeneous mapping34

Hutchinson operator27, 28

I

implicit function theorem57

infinitesimal generator49, 50

initial value problem44, 45

inner product5

isometry7, 8

isomorphism8

iterated function system25, 27

J

Jacobian matrix29

K

Krein-Rutman39

Krein-Rutman theorem30

L

Lebesgue spaces8

limit point3

linear functional8

Lipschitz mapping15

local Lipschitz condition45

M

measure zero9

metric3

metric space3

mild solution49, 50

mild solutions44

minimizing sequence62

monotone34, 72

monotone mapping21

monotone norm37

multiindex6

88


N

Nagumo’s lemma78

Newton iteration scheme29

Newton’s method29

Newton-Raphson method29

nonexpansive15

norm4

norm topology62

normed vector space4

O

obstacle problem67

open ball3

open cover3

open set3

oscillation31

P

pairing62

partial order30

Perron-Frobenius39

Perron-Frobenius theorem30

Picard iteration46

Picard-Lindelof theorem45

Poincare’s inequality70

Poincare68

positive eigenvectors30

positive mapping34

positive matrix30

projective diameter35

projective diameter40

pseudo-metric7

Q

quasilinear81

quasiperiodic55

R

reflexive Banach space62

resolvent set73

Riemann integral8

S

self-adjoint73

self-similar28

semigroup49, 50

semilinear69

Sierpinski triangle28

similarity transformation28

similarity transformations28

Sobolev space11, 67

solid cone30

spectral radius73

spectrum73

strongly continuous49, 50

sup norm5

support8

symmetric62, 73

T

Taylor expansion57

totally bounded3, 15triangle inequality3

Uuniformly positive39

Vvariational inequality63, 65

W

weak convergence62weak topology62

weakly closed62

weakly lower semicontinuous62


Robert M. Brooks

Department of Mathematics, University of Utah, Salt Lake City, UT 84112, USA

E-mail address: [email protected]

Klaus Schmitt

Department of Mathematics, University of Utah, Salt Lake City, UT 84112, USAE-mail address: [email protected]

the contraction mapping principle and some … · a metric space m is said to be complete if, and...

Documents